Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Martens, Adam
Doctor of Philosophy - PhD
University of British Columbia
In this dissertation, we study the Ricci flow and its applications on manifolds that have small enough scale-invariant integral curvature (so-called "curvature concentration") relative to its Sobolev bounds. Our primary results are summarized in the following three items. (1) We show that if the bounds are assumed to be global for a bounded curvature manifold, then the Ricci flow emanating from this initial data exists for all time, with faster than critical curvature decay. We also show topological rigidity assuming that the curvature is globally sub-critically integrable - an assumption that is, for example, satisfied by the well-studied family of "asymptotically flat" manifolds. (2) We sharpen a gap theorem of Chan and Lee for nonnegative Ricci curvature manifolds (possibly unbounded curvature) that have positive asymptotic volume ratio (in the nonnegative Ricci curvature setting, we show that volume bounds are equivalent to Sobolev bounds) and small enough curvature concentration, by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than critical curvature decay (item (1) above is used as one element of the proof), which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap. (3) In recent work of Chan-Huang-Lee, it is shown that if a (possibly unbounded curvature) manifold enjoys uniform bounds on the negative part of the scalar curvature, the local entropy, and volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially. We show that the bound on scalar curvature assumption above is redundant. We also give some applications of this quantitative short-time existence, including Ricci flow smoothing for measure space limits, a Gromov-Hausdorff compactness result, and topological/geometric rigidity in the case that the a priori local bounds are strengthened to be global.
Ricci flow: curvature concentrationand Sobolev boundsbyAdam MartensB.Sc., The University of British Columbia, 2019M.Sc., The University of British Columbia, 2021A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2025© Adam Martens 2025The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Ricci flow: curvature concentration and Sobolev boundssubmitted by Adam Martens in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Mathematics.Examining Committee:Albert Chau, Professor, Mathematics, UBCSupervisorAilana Fraser, Professor, Mathematics, UBCSupervisory Committee MemberSe´bastien Picard, Assistant Professor, Mathematics, UBCSupervisory Committee MemberStephen Gustafson, Professor, Mathematics, UBCUniversity ExaminerMark Van Raamsdonk, Professor, Physics and Astronomy, UBCUniversity ExaminerRichard Bamler, Professor, Mathematics, University of California, BerkeleyExternal ExamineriiAbstractIn this dissertation, we study the Ricci flow and its applications on man-ifolds that have small enough scale-invariant integral curvature (so-called“curvature concentration”) relative to its Sobolev bounds. Our primaryresults are summarized in the following three items.(1) We show that if the bounds are assumed to be global for a boundedcurvature manifold, then the Ricci flow emanating from this initial dataexists for all time, with faster than critical curvature decay. We also showtopological rigidity assuming that the curvature is globally sub-criticallyintegrable - an assumption that is, for example, satisfied by the well-studiedfamily of “asymptotically flat” manifolds.(2) We sharpen a gap theorem of Chan and Lee for nonnegative Ricci cur-vature manifolds (possibly unbounded curvature) that have positive asymp-totic volume ratio (in the nonnegative Ricci curvature setting, we show thatvolume bounds are equivalent to Sobolev bounds) and small enough curva-ture concentration, by showing that the curvature concentration need onlydepend linearly on the asymptotic volume ratio. We prove the result byexhibiting a long-time Ricci flow solution with faster than critical curvaturedecay (item (1) above is used as one element of the proof), which allows usto shift the limiting contradiction argument to time infinity and thus obtainan explicit bound on the size of the gap.(3) In recent work of Chan-Huang-Lee, it is shown that if a (possiblyunbounded curvature) manifold enjoys uniform bounds on the negative partof the scalar curvature, the local entropy, and volume ratios up to a fixedscale, then there exists a Ricci flow for some definite time with estimates onthe solution assuming that the local curvature concentration is small enoughinitially. We show that the bound on scalar curvature assumption aboveis redundant. We also give some applications of this quantitative short-time existence, including Ricci flow smoothing for measure space limits, aGromov-Hausdorff compactness result, and topological/geometric rigidity inthe case that the a priori local bounds are strengthened to be global.iiiLay SummaryA common theme in modern mathematics is the extent to which anobject that has irregularities can be smoothed out into a more desirableone. We study one method of smoothing called the Ricci flow, under theassumptions that the degree to which the object is considered not ‘flat’ (ina particular averaged sense) is small enough and that it is not too irregular.In this case, we show that the object can indeed be smoothed out - atleast a little bit - by the Ricci flow in a controlled way. Moreover, if theaforementioned initial control is known to hold in a cumulative sense (thatis, across the whole object), then we can actually make conclusions on whatit must have looked like initially. In some settings, these conclusions aresufficient to actually classify precisely what the initial object was.ivPrefaceThis dissertation is a culmination of three papers, two of which I am thesole author [34], [33], and the other is joint work with Albert Chau [7].Chapter 1 is mostly expository wherein we state the main results (The-orems A - G) and describe their relation to each other, as well as any back-ground knowledge necessary.The majority of Chapter 2, with the exception of Section 2.5, is a sum-mary and slight strengthening of known results in the literature, and manyof the results in this section appear in some form or another in each of[7], [34], [33]. Section 2.5 is adapted from [34], and is original intellectualproperty of myself.Chapter 3 presents the proofs of Theorems A and B, which originallyappear in [7]. This paper, entitled Long-time Ricci flow existence and topo-logical rigidity from manifolds with pinched scale-invariant integral curva-ture, has been submitted for publication and is joint with Albert Chau.Though these are joint work, the original ideas and implementation of thework presented in this chapter were my own with the exception of Section3.1, Proposition 3.2.11, and the proof of Theorem B, each of which werecontributed to equally by myself and Albert Chau.Chapter 4 presents the proof of Theorems C and D, which originallyappear in [34]. This paper, entitled Sharpening a gap theorem: nonnegativeRicci and small curvature concentration, has been published in the journalCalculus of Variations and Partial Differential Equations (volume 64, articlenumber 79, 2025), and I am the sole author. Unless otherwise stated, thework in this chapter is original intellectual property of myself.Chapter 5 presents the proof of Theorems E, F, and G, which originallyappear in [33]. This paper, entitled Removing scalar curvature assump-tion for Ricci flow smoothing, has been published in the journal Bulletin ofthe London Mathematics Society (https://doi.org/10.1112/blms.70073)vand I am the sole author. Unless otherwise stated, the work in this chapteris original intellectual property of myself.Generative AI was not used for any aspect of the work within this dis-sertation.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Long-time existence of the flow . . . . . . . . . . . . . . . . . 41.3 A gap theorem: nonnegative Ricci curvature . . . . . . . . . 91.4 Removing scalar curvature assumption . . . . . . . . . . . . 161.5 Notation and conventions . . . . . . . . . . . . . . . . . . . . 232 Sobolev Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1 Perelman’s localized µ- and ν-functionals . . . . . . . . . . . 272.2 ν-functional from Sobolev inequality . . . . . . . . . . . . . . 302.3 Sobolev inequality from ν-functional . . . . . . . . . . . . . . 322.4 Sobolev inequality along the flow . . . . . . . . . . . . . . . . 342.5 Sobolev inequality in nonnegative Ricci curvature . . . . . . 363 Long-Time Existence . . . . . . . . . . . . . . . . . . . . . . . 443.1 Upper continuity of the Lp-curvature . . . . . . . . . . . . . 453.2 Long-time existence and convergence . . . . . . . . . . . . . 503.3 Unbounded curvature initial data . . . . . . . . . . . . . . . 733.4 An explicit bound for L(n) . . . . . . . . . . . . . . . . . . . 76vii3.4.1 Modified Marcinkiewicz . . . . . . . . . . . . . . . . . 793.5 Wrinkled balls . . . . . . . . . . . . . . . . . . . . . . . . . . 814 A Gap Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1 Proof of Theorem C from Theorem D . . . . . . . . . . . . . 864.2 Evolution of distance . . . . . . . . . . . . . . . . . . . . . . 874.3 Evolution of curvature concentration . . . . . . . . . . . . . 904.4 A bounded curvature localized result . . . . . . . . . . . . . 924.5 Proof of Theorem D . . . . . . . . . . . . . . . . . . . . . . . 1045 Removing Scalar Curvature Dependence . . . . . . . . . . . 1135.1 Modified distance distortion . . . . . . . . . . . . . . . . . . 1135.2 Localized Ricci flow smoothing . . . . . . . . . . . . . . . . . 1205.3 Smoothing of measures . . . . . . . . . . . . . . . . . . . . . 1255.3.1 Existence and properties of d0 . . . . . . . . . . . . . 1275.3.2 Existence and properties of µ . . . . . . . . . . . . . . 1305.4 Topological and volume rigidity . . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146viiiList of Figures1.1 Smoothing a deformed sphere [18] . . . . . . . . . . . . . . . 11.2 Contradiction is derived in limiting space . . . . . . . . . . . 91.3 Theorem C gives a linear lower bound (red) for f(v) . . . . . 122.1 Nonnegative Ricci completes the circle of equivalences . . . . 362.2 Fitting in a small ball along a geodesic ray . . . . . . . . . . . 393.1 A wrinkle is added to the interior of Euclidean ball . . . . . . 834.1 Approximating Lengthg(τ)(γ) from unit g(τ)-balls . . . . . . . 89ixAcknowledgmentsFirst of all, I would like to express my sincere thanks to Albert for beingmy supervisor. I fondly remember the first differential geometry class thatyou taught me, our countless discussions, and working with you over thecourse of several papers. You have been gracious in your understanding andgenerous with your time. I consider myself enormously blessed to have beenyour student. Enjoy the ‘carfire’.Also, very special thanks to Gordon Slade for endorsing and believing in myteaching abilities, and just for being such a pleasure to work with. I wishyou all the best in your very long sabbatical!I unfortunately don’t have room to thank everyone who has helped methrough this degree or who has made a lasting impact on me, but I wouldlike to specifically thank the following people:The examining committee, for their time and thoughtful questions;Richard Bamler, for his kind and helpful comments on this dissertation;Brian Marcus, who has helped me in countless ways and is a true friend;Joshua Zahl, for being an exemplary teacher;Richard Anstee, whose door was always open;Rachel Olivier, for her advice that I’ll never forget;Fok-Shuen Leung, for his feedback and confidence in my teaching; andAndrew Rechnitzer, for all his help and guidance (and bullshit).Thank you also to all the exceptional and inspiring teachers that I havelearned so much from, including (but certainly not limited to) Dragos Ghioca,Malabika Pramanik, Zachary Hudson, as well as all those previously men-tioned. I look up to all of you.I also greatly appreciate all those students (a particular shout-out to theOG 120/121 crew!) who have encouraged me to pursue teaching; know-ing that I have made a difference in their education is what motivates meprofessionally.Finally, my deepest gratitude goes to my wife Stephanie, who promised toxproofread for me (but never did), and to my children, without whom I wouldhave finished my degree much earlier. (love you)PS. Thanks Dan for suggesting that I apply for the UBC postdoc! I’m look-ing forward to sticking around a little while longer.xiTo the world we dream about! And the one we live in now.-Livin’ it Up on Top, HadestownxiiChapter 1Introduction1.1 BackgroundOne of the most common themes in modern day mathematics is the ideaof “smoothing” an object that may have unwelcome irregularities into amore desirable one. Imagine, for instance, a crumpled sheet of paper slowlyunfolding itself back into a flat one, or a ‘kicked in’ soccer ball reshapingitself to become round again as in, for example, Figure 1.1.Figure 1.1: Smoothing a deformed sphere [18]The question now is: How do we take this intuitive understanding of“smoothing”, and use it to write down an evolution equation that can beapplied to abstract, higher-dimensional shapes, i.e., manifolds.Well, if we look back at the illustrative smoothing ball Figure 1.1, we cansee that the way in which this ball is smoothed out is that the larger ‘round’components decrease in size over time, while the smaller ‘neck’ componentsincrease in size. Together these have a homogenizing effect. Translating thisinto Riemannian geometry jargon, we know that the ‘round’ components arepositively curved, while the ‘neck’ components are negatively curved. So ifwe were trying to write down a smoothing equation for manifolds, we mightwant something like∂∂tg(t) = −[sign of the curvature].1Here g(t) is the Riemannian metric of the resulting shape at time t along theevolution. Since the left-hand side of this evolution equation is a symmetric2-tensor, the right-hand side should be as well. So then the question is,“what is a symmetric 2-tensor that also represents curvature?”. One sim-ple answer, of course, is the Ricci curvature, Rcg(t). So for this evolutionequation to make sense, we ought to take something more like∂∂tg(t) = −Rcg(t).There are other options of course, but it turns out that this choice is simpleenough for us to work with, but complicated enough that it is mathemati-cally useful. Our last modification of this evolution equation is to introducea constant factor 2 in the right-hand side, just for convenience in subsequentcalculations. The result is the Ricci flow, introduced by Hamilton [22] in1982, which deforms an initial Riemannian metric g0 on a n-dimensionalmanifold Mn according to the equation∂∂tg(t) = −2Rcg(t),g(0) = g0.(1.1)It was proved in [22] that when the initial manifold (Mn, g0) is compact,then the Ricci flow (1.1) exhibits a unique smooth solution g(t) on M ×[0, T (g)). Following Hamilton, a classical result of Shi [39] says that if g0is complete with sectional curvatures uniformly bounded by K, then (1.1)admits a complete bounded curvature solution g(t) onMn× [0, T (n,K)) forsome time T (n,K) > 0 depending only on n and K, and in no other way onthe metric g0. Extending Shi’s theorem to cases of unbounded curvature, orby extending the solution for all times t ∈ [0,∞) under suitable conditionsare of fundamental importance in Ricci flow theory and applications.One of the central themes in the study of Ricci flow is determining whichinitial manifolds (M, g) be deformed into the standard flat space form, as-suming that it satisfies some “almost-Euclidean” forcing condition. Thealmost-Euclidean condition that we will be considering throughout is small-ness of the curvature concentration:∫M|Rm|n2g dVg.Note that the power n2 here is the critical one in the sense that it is the onlypower p such that the integral curvature ∥|Rm|∥Lp is scale-invariant. Thus,2the curvature concentration can be thought of as describing some intrinsicproperty of the manifold and is the clear choice of exponent if one is lookingfor scale-invariant conclusions as well (such as, say, topological rigidity).Regularization of a manifold assuming small curvature concentration hasbeen extensively studied [31, 44–48] where, in addition to an assumption onthe curvature concentration, there is also an assumption on the non-scaling-invariant Lp-norm of the Ricci curvature ∥Rc∥Lp , p > n2 . In particular, theboundedness of ∥Rc∥Lp allows one to obtain a point-wise bound on Rcg(t)for later times t > 0 along the flow, resulting in a C0-comparison betweeng(0) and g(t). So if one wishes to eliminate the assumption about the non-scaling-invariant integral curvature quantity ∥Rc∥Lp , then it seems prudentto assume some other C0-condition that is (semi) preserved along the flow.The C0-initial condition that we will be primarily assuming as a hypoth-esis is an L2-Sobolev inequality(∫u2nn−2 dVg)n−2n≤ Cg∫|∇gu|2g dVg, (1.2)for all compactly supported W 1,2 functions.This choice of C0-condition turns out to be a very convenient one in theRicci flow setting, when it is paired with an assumption of smallness of thecurvature concentration, because, in this setting, the L2-Sobolev inequality(1.2) turns out to be (semi) preserved along the flow. Let’s take a momentto explain this. The first step is to show that L2-Sobolev inequalities areequivalent to control on the ν¯-functional (see Section 2.1 for the definitionand Sections 2.2 and 2.3 for the equivalence). Then if the curvature concen-tration is small enough compared to the L2-Sobolev constant Cg, specificallyif (∫|Rm|n2g dVg) 2n≲ 1Cg, (1.3)then we can translate back-and-forth between control on the ν¯-functionalto the more classical ν-functional. The way this is accomplished is as fol-lows. Recall (or reference Section 2.1), the only difference between thesefunctionals is the presence of an extra weighted scalar curvature termτ∫Rgu2 dVg.3By applying Ho¨lder’s inequality, the assumed L2-Sobolev inequality (1.2),and our smallness condition on the curvature concentration as in (1.3), weobtainτ∣∣∣∣∫ Rgu2 dVg∣∣∣∣ ≲ τ (∫ |Rm|n2g dVg) 2n (∫ u 2nn−2 dVg)n−2n≲ τ 1CgCg∫|∇gu|2g dVg = τ∫|∇gu|2g dVg.Because the right-hand side above is the same as a term in the definition ofthe W -entropy, this piece can be reabsorbed at the expense of a constant.In the sense as described above, the smallness condition on the curvatureconcentration relative to the inverse of the L2-Sobolev constant as in (1.3)is the “correct” one, at least in view of the parabolic Ricci flow approachand Ho¨lder’s inequality. Indeed, this is precisely the condition that we willbe working with most of the time in this work.We end this background section by noting that both the L2-Sobolevinequality (1.2) and the curvature concentration condition (1.3) are scale-invariant, so it should be expected that, as long as these are the only con-ditions that we will be assuming, then our conclusions should also be scale-invariant as well. In particular, this is suggestive - if these properties areassumed to hold globally - that a Ricci flow emanating from such a manifoldmay in fact be a long-time flow, i.e., a flow which exists for all t ∈ [0,∞).Such flows are particularly powerful when coupled with curvature decay thatis faster than the scale-invariant decay rate of|Rm|g(t) ≤Ct.Such conclusions together lead to rigidity statements on the topology andvolume growth of the manifold, because one can obtain that the tangent cone(the result of successive blow-downs) of g(0) exhibits a Ricci flow solutionthat is necessarily the static flow on flat Euclidean space. This is a recurringtheme in the proofs of our main results.1.2 Long-time existence of the flowOur first main theorem generalizes the following result of Chen, whichgives long-time Ricci flow existence and metric convergence (and thus also4topological rigidity) when (M, g0) satisfies conditions (1.2) and (1.3) as wellas the (somewhat stringent) condition that g is asymptotically flat. Thelatter condition is that for some compact set K ⊂ M , M \K is diffeomor-phic to Rn \BgE (1) and that, in this coordinate system, the metric and itsderivatives decay like|∂(α)(gij − δij)| ≲ r−k−τ ,where r is the Euclidean distance, α is a multi index of order k, and τ iscalled the asymptotic order of g. We direct the reader to [13] for the precisedefinition of asymptotic flatness that is being used in the following.Theorem 1.2.1. [13, Theorem A] For n ≥ 3, there exists a dimensionalconstant δ(n) > 0 such that the following holds. Let (Mn, g) be an asymp-totically flat manifold of order τ > 0. Suppose that (Mn, g) satisfies theSobolev inequality(∫Mu2nn−2 dVg)n−2n≤ Cg∫M|∇gu|2g dVgfor all u ∈W 1,20 (M), and the scale-invariant curvature pinching(∫M|Rm|n2g dVg) 2n≤ δ(n) 1Cg.Then the complete Ricci flow g(t) with initial condition g(0) = g exists forall times t ∈ [0,∞) and satisfieslimt→∞ t supx∈M|Rm|(x, t) = 0.In particular, g(t) converges in C∞−τ ′(M) for any τ′ ∈ (0,min(τ, n − 2)) toRn with the Euclidean metric and thus Mn is diffeomorphic to Rn.The first of our results is to generalize Theorem 1.2.1 to complete non-compact bounded curvature manifolds, by removing the ‘asymptotically flat’condition. We prove the following. Note in particular the theme of scale-invariance of the hypotheses and conclusions.Theorem A. For n ≥ 3, there are dimensional constants δ(n) > 0 andC0(n) = (5e)n2 such that the following holds. Let (Mn, g) be a complete5bounded curvature manifold. Suppose that (Mn, g) satisfies the Sobolev in-equality (∫Mu2nn−2 dVg)n−2n≤ Cg∫M|∇gu|2g dVg (1.4)for all u ∈W 1,20 (M), and the scale-invariant curvature pinching(∫M|Rm|n2g dVg) 2n≤ δ(n) 1Cg. (1.5)Then the complete Ricci flow g(t) with initial condition g(0) = g exists forall times t ∈ [0,∞) and satisfies the following curvature estimates:(a) For all t ∈ (0,∞) there holdssupx∈M|Rm|(x, t) ≤ C0(n)t.(b) The singularity at infinity satisfieslimt→∞ t supx∈M|Rm|(x, t) = 0.Moreover, if |Rm|g ∈ Lp(M) for some p < n2 , then g(t) converges locallysmoothly to a complete flat limit metric g(∞) on Mn with Euclidean volumegrowth, and thus Mn is diffeomorphic to Rn.Remark 1.2.2. The constant δ(n) in Theorem A can be taken to beδ(n) ={10−6 n = 315000n2n ≥ 4.Here we have used the estimates on L(n) (this is the dimensional constantfrom Theorem 2.4.1) in (3.34) as well as the requirements on the size of δin Section 3.2 thatδ(n) <140nL(n)(1.6)The reason why we provide a different bound on δ(n) for n ≥ 4 is to obtain abetter estimate on the ϵ(n) in Theorem C, which is only stated for dimensionsn ≥ 4. It is also worth noting here that δ(n) may be far from sharp; it isan interesting question precisely how small δ need be for the conclusions ofTheorem A to continue to hold.6Remark 1.2.3. When (Mn, g) is asymptotically flat, the condition that|Rm| ∈ Lp for some p < n2 is automatically satisfied whenever p > n2+τ whereτ > 0 is the order of the asymptotic end. Our proof of Proposition 3.2.11 (see(3.26) and (3.27)) establishes that for any σ < τ/2, we have the global decayestimate |∇kRm| ≲ t−1−σ−k/2. Using this, it follows from [32, Theorem 5.1]that in fact g(t) converges in C∞−τ ′(Mn) for any τ ′ ∈ (0,min(τ, n− 2)) to Rnwith the Euclidean metric, in the same sense as Theorem A. Thus, TheoremA truly generalizes Theorem 1.2.1.One of the critical downsides of Theorem A is the assumption that theinitial manifold has bounded curvature. Though in application we are oftenable to work around this by applying it to a metric at a later time along theflow, this is somewhat unsatisfactory. This bounded curvature assumptionalso seems like it should be superfluous due to its function as a stepping stoneonly; getting the flow started, knowing that the global integral curvaturedoesn’t instantly blow up, applying maximum principle, etc, all use the factthat the initial manifold has bounded curvature.The following theorem partly makes up for this shortcoming by showingthat the same result of Theorem A holds as long as the initial manifold is asmooth limit of such bounded curvature manifolds.Theorem B. With n ≥ 3 and δ(n), C0(n) as in Theorem A, the follow-ing holds. Suppose that (Mn, g) is a complete Riemannian manifold (notnecessarily bounded curvature) which is the pointed smooth local limit ofa sequence of complete bounded curvature Riemannian manifolds (Mi, gi)(i = 1, 2, . . . ) such that for each i ∈ N, there holds:(i) The Sobolev inequality(∫u2nn−2 dVgi)n−2n≤ Cgi∫|∇giu|2gi dVgi ; and(ii) Sufficiently pinched scale-invariant integral curvature conditionssupi∈N(∫M|Rm|n2gi dVgi) 2n≤ δ(n)supi∈NCgi.Then there exists a complete solution g(t) to the Ricci flow on M × [0,∞)with g(0) = g that satisfies the curvature estimate7supx∈M|Rm|(x, t) ≤ C0(n)tfor all t ∈ (0,∞). Moreover if supi∈N∥|Rm|gi∥Lp(gi) <∞ for some p < n2 , theng(t) converges locally smoothly to a complete flat limit metric g(∞) on Mnwith Euclidean volume growth, and thus Mn is diffeomorphic to Rn.Although the statement of Theorem B is quite satisfactory, its usefulnessin practice is limited due to its external nature. What we mean by this isthat the hypotheses require the a priori existence of a sequence of boundedcurvature metrics with the property that each one has small global curva-ture concentration. This is a problem because in practice, one is given anunbounded curvature metric and would like a way to approximate it withsuch bounded curvature manifolds, but any known method of approxima-tion (“blowing up the boundary” [25], for example) results in the globalcurvature concentration of the approximating sequence being large.This ‘external nature’ of Theorem B is partially rectified later whenwe discuss localized Ricci flow smoothing (see Theorems E and 4.4.2) thatonly assume local control on the curvature concentration and local Sobolevinequalities, because these local quantities can be forced to be semi-preservedwhen approximating a given unbounded curvature manifold by pushing theboundary to infinity. See Section 4.5 for more detail.Remark 1.2.4. The hypotheses of Theorem A impose no restriction onthe supremum of, or decay/growth rate of the curvature. In fact, givenany sequence of positive numbers {αi} ⊂ R+, one can construct a completemetric g on Rn such thatsupBgE (i+1)\BgE (i)|Rm|g = αibut for which g satisfies the Sobolev inequality (1.4) and the integral curva-ture pinching estimate (1.5) with Cg arbitrarily close to CgE . In particular,one can define a sequence of complete bounded curvature metrics gi on Rnthat converge locally smoothly to an unbounded curvature metric g on Rnsuch that the metrics gi satisfy the hypotheses (i) and (ii) from Theorem B.The described constructions can be accomplished by gluing into the flat Eu-clidean space (Rn, gE) countably many “wrinkled balls”, by which we meana compact perturbation h of the Euclidean metric on the open unit ball Bwhich satisfies:81. (1 + c1)−1gE ≤ h ≤ (1 + c1)gE on B,2. |Rm|h(x) > c2 for some x ∈ B, and3.∫B |Rm|n2h dVh < c3,where c1, c2, c3 > 0 can be prescribed arbitrarily. See Section 3.5 for thisconstruction.1.3 A gap theorem: nonnegative Ricci curvatureA central aspect of Ricci flow analysis is the tendency for the flow toroughly preserve the geometry of almost Euclidean regions, despite the man-ifold’s global geometry being possibly not well understood. This is the so-called “pseudolocality” of the Ricci flow. Although the sense in which aregion is “almost Euclidean” may differ, the general arguments of pseudolo-cality theorems often follow this outline:1. Construct a sequence of Ricci flows which are becoming more “locallyEuclidean” but for which some undesirable behaviour occurs (often ofthe form of large curvature).2. Normalize the metrics to focus on the local region where the undesir-able behaviour occurs.3. Analyze the sequence of normalized metrics as they approach a limitingspace, wherein a contradiction is apparent.We will refer to this method of proof (where a contradiction is derived inthe limiting space) as a “limiting contradiction argument”. see Figure 1.2below.Figure 1.2: Contradiction is derived in limiting space9A notable aspect of such arguments is that they usually involve constantsthat are difficult or impossible to provide numerical bounds for due to thenature of the contradiction appearing in the limit space instead of at aspecific moment along the sequence.One of the reasons why pseudolocality theorems are important is thatthey relay geometric and topological information in the local region. Inparticular, if the sense in which a manifold M is “almost Euclidean” isinvariant under scaling and is satisfied everywhere onM , then one can applythe result to a sequence of blow-downs of the initial metric, thus providinginformation about the manifolds global geometric structure. Consider forexample the following notable result of Chan & Lee.Theorem 1.3.1. [6, Theorem 1.1] For any n ≥ 4 and v > 0 there ex-ists some ϵ, depending on n and v, such that if (Mn, g) is complete withnonnegative Ricci curvature Rcg ≥ 0, asymptotic volume ratio satisfyinglimr→∞VolgBg(x0, r)ωnrn≥ v,and globally small curvature concentration∫M|Rm|n2g dVg ≤ ϵ(n, v), (1.7)then (Mn, g) is isometric to (Rn, gE).The proof given in [6] is a limiting contradiction argument wherein theyfix some v > 0 and then assume there is a sequence of nonnegative Riccicurved (Mni , gi), each with asymptotic volume ratio at least v and integralcurvatures satisfying (1.7) with ϵi → 0, which results in a contradiction ina limiting space. This raises two questions:1. For any fixed v > 0, precisely how small does ϵ(v) need to be to liewithin the desired gap?2. Given two distinct v1, v2 > 0, how does ϵ(v1) relate to ϵ(v2)?The main result of this section provides an answer to both of thesequestions. We prove the following.Theorem C. For n ≥ 4, there exists a dimensional constant ϵ(n) ≥ (10n)−10n3such that the following holds. Let (Mn, g) be any complete Riemannian man-ifold (not necessarily bounded curvature) with nonnegative Ricci curvatureRcg ≥ 0. Suppose that (Mn, g) has positive asymptotic volume ratio10limr→∞VolgBg(x0, r)ωnrn≥ v > 0,and curvature concentration satisfying∫M|Rm|n2g dVg ≤ ϵ(n)v. (1.8)Then (Mn, g) is isometric to (Rn, gE).Even though Theorem C is stated for dimensions n ≥ 4, there is nogenerality lost with this assumption because it follows from work of Cheeger[9, Theorem 6.10] and Colding’s Volume Continuity Theorem [17] that forany odd n (in particular, n = 3), merely the assumption of ∥Rm∥Ln2 (g)<∞is sufficient to guarantee that any nonnegative Ricci curvature manifold withEuclidean volume growth is necessarily isometric to (Rn, gE).When n is even, the picture is less clear. For instance, we have the exam-ple of the Eguchi-Hanson metric which is a nonflat 4-dimensional Ricci flatmanifold with uniformly bounded from below volume ratios (on all scales)and ∥|Rm|∥Ln2< ∞. For the dimensions n in which such examples exist,there is a well-defined function fn : (0, 1] → R which defines the “size ofthe gap” for manifolds with at least the given asymptotic volume ratio, i.e.,fn(v) is the supremum of all ρ > 0 for which{(Mn, g) with Rcg ≥ 0, AVR(g) ≥ v and∫M|Rm|n2g dVg ≤ ρ}= {(Rn, gE)},where both of the sets of Riemannian manifolds above is understood in thesense of equivalent classes modulo isomorphism.The general behaviour of fn(v) is currently not well understood. Forinstance, it is clear that v 7→ fn(v) is nondecreasing, but it is not known (tothe author’s knowledge) whether fn is uniformly bounded from below or iffn(v)→ 0 as v → 0. In light of Theorem C however, which provides a linearlower bound for f(v), we have a very natural follow-up question of whetheror not f(v) must also be squeezed underneath some linear larger line (green).Thus we have the following picture (Figure 1.3) of some possibilities for whatf(v) could look like, with the result of Theorem C indicating a lower boundfor f (in red).11Figure 1.3: Theorem C gives a linear lower bound (red) for f(v)In other words, we can ask whether or not the linear relationship betweenthe required smallness of the curvature concentration and the asymptoticvolume ratio v in Theorem C is sharp with respect to the asymptotic volumev? More precisely, we have the following question.Question 1.3.2. Is it true thatlim supv→0+fn(v)v<∞?Richard Bamler [1] has indicated that his intuitive belief is that it is im-possible for Question 1.3.2 to have an affirmative answer, in fact he suggeststhat fn likely must remain uniformly bounded from below as v → 0. His in-tuition as to why this may be the case is that the joint smallness of AVR(g)and ∥|Rm|g∥Ln/2 is likely to force the Euler number and all the Pontryaginnumbers of such a manifold to vanish, suggesting that the manifold is justflat Euclidean space.I am inclined to agree with Bamler’s intuition. On the other hand, I amalso obliged to mention here that there is at least a perceived sharpness ofTheorem C due to the fact that its method of proof is sharp in the sensethat if we assumed any worser than linear relationship to v, then the proofwould no longer go through (at least with the current Ricci flow machinery).The reason for this is as follows. Recall the relationship between the curva-ture concentration and an L2-Sobolev inequality via Ho¨lder’s inequality as12discussed in Section 1.1. In particular,∫M|Rm|gu2 ≤(∫M|Rm|n2g) 2n(∫Mu2nn−2)n−2n≲ v 2n(∫Mu2nn−2)n−2n.This coefficient of v2n on the right-hand side turns out to be the largest onepossible that still allows the proof to run because our translations betweenvolume ratios and Sobolev constants is sharp with respect to v (see Chap-ter 2) and the particular power of 2n is precisely the one that is necessaryto eliminate the scalar curvature term in the ν-functional as well as the|Rm|n2+1 term in the evolution of |Rm|n2 along the Ricci flow (see the proofof Theorem D in Chapter 4). This is suggestive that the linear relationshipof curvature concentration to v as in (1.8) is the largest one that the Ricciflow approach can handle.The way in which we are able to obtain an explicit bound on ϵ(n) is todemonstrate the existence of a long-time solution to the Ricci flow with fasterthan 1/t curvature decay, which allows us to shift the limiting contradictionargument from the curvature concentrations bound to time infinity of theaforementioned long-time Ricci flow. It is worth noting in the followingresult that we obtain no control on the rate of which t|Rm|g(t) → 0. Thisis the sacrifice we make in exchange for obtaining an explicit bound on theconstant ϵ(n) in the statement of Theorem C.Theorem D. For any n ≥ 4 and v > 0, there exists c(n) > 0, ι(n, v) > 0such that the following holds. Suppose that (Mn, g) satisfies the hypothesesof Theorem C with this choice of n, v. Then there exists a long-time solutiong(t), t ∈ [0,∞) to the Ricci flow with g(0) = g which satisfies:(i) Rg(t) ≥ 0 for every t ∈ [0,∞);(ii) |Rm|g(t) ≤ 1t for every t ∈ (0,∞);(iii) injg(t) ≥ ι(n, v)√t for every t ∈ (0,∞);(iv) Volg(t)Bg(t)(x, r) ≥ c(n)v rn for every x ∈M , r > 0, t ∈ (0,∞); and(v) limt→∞ t|Rm|g(t) = 0.The proof of Theorem C follows directly from Theorem D and distancedistortion along the Ricci flow, see Section 4.1 for the details. Unsurprisinglyin light of the presence of conclusion (v), Theorem A is instrumental in the13proof of Theorem D. What may be surprising however is that Theorem Acan actually be used several times (not just at the end to give conclusion(v)), see the proof of Step 2 in Theorem 4.4.2, for example.If we knew in advance that (Mn, g) had bounded curvature, we couldcircumvent the need for Theorem D in the proof of Theorem C by directlyapplying Theorem A. In particular, by Corollary 2.5.8, we could obtain theSobolev inequality(∫M|u| 2nn−2 dVg)n−2n≤ C(n)v− 2n∫M|∇gu|2g dVgfor all compactly supported u ∈ W 1,2(M). If we restrict our ϵ(n) smallenough so thatϵ(n) ≤(δ(n)C(n))n2where δ(n) is the dimensional constant in Theorem A, then by the conclusionof that result, we know that the complete bounded curvature Ricci flow g(t)with initial condition g(0) = g, exists for all times t ∈ [0,∞) and satisfiesthe curvature estimatessupx∈M|Rm|g(t)(x) ≤C0(n)t,for any t ∈ (0,∞), andlimt→∞ t supx∈M|Rm|g(t)(x) = 0.This gives conditions akin to (ii) and (v). Moreover, by Corollary 3.2.3 (thiscorollary is part of the proof of Theorem A) and Lemma 2.5.1, we know thatthere exists a v0(n, v) > 0 such thatVolg(t)Bg(t)(x, r) ≥ v0(n, v)rn for any t ≥ 0, r > 0, x ∈M,which gives (iv). Penultimately, for any t > 0, the conditionsVolg(t)Bg(t)(x,√t) ≥ v0tn2 and |Rm|g(t) ≤C0t,together imply the injectivity radius boundinjg(t)(x) ≥ ι(n, v0)√t14by [10] for some ι(n, v0) > 0 which does not depend on t > 0. This givescondition (iii). Finally, condition (i) follows immediately from the assump-tion that g(t) is a bounded curvature flow (so in particular, lower scalarcurvature bounds are preserved by the maximum principle). The proof ofTheorem C then follows synonymously (see Section 4.1).The issue that arises when (M, g) has unbounded curvature is not estab-lishing short-time existence of the flow. Indeed, by [5, Theorem 4.1], we cansay that if (M, g) satisfies the hypotheses of Theorem C (for some v > 0),then for any α > 0 there is a Ricci flow g(t) with g(0) = g, defined on someshort time interval [0, T (α, g)], which enjoys α/t curvature control. Rather,the issues that we primarily face in this case are:1. Extending the flow for all times t ∈ [0,∞).2. Establishing the faster than 1/t curvature decay as in (v) in TheoremD.We get around both of these issues by showing that the unique Ricciflow g(t) with 1/t curvature control1 exists for all time and preserves theSobolev inequality and curvature concentration of the initial metric (up todimensional constants). Let us explain this in slightly more detail: Just asmentioned in the bounded curvature case above, we obtain from an asymp-totic volume ratio at least v > 0, the Sobolev inequality(∫M|u| 2nn−2 dVg)n−2n≤ C(n)v− 2n∫M|∇gu|2g dVg.The Ricci flow that we construct will satisfy the properties|Rm|g(t) ≤ 1t ,∫M |Rm|n2g(t) dVg(t) ≤ C1(n)ϵ(n)v, and(∫M |u|2nn−2 dVg(t))n−2n ≤ C1(n)C(n)v− 2n∫M |∇g(t)u|2g(t) dVg(t).(1.9)This preservation of the Sobolev inequality and smallness of the curva-ture concentration allows us to apply Theorem A with the bounded curva-ture metric g(1) replacing g(0), from which we obtain the faster than 1/tcurvature decay.1Uniqueness of Ricci flows with c/t curvature control was recently announced by Lee[28], and the existence of such a flow for some short time is guaranteed by [5].151.4 Removing scalar curvature assumptionAs a stepping stone in our proof of Theorem C, we require the use ofa quantitative short-time existence result Theorem 4.4.2, which has vari-able localized hypotheses (see the hypothesis (4.3)). If the hypotheses arenot variable, then this quantitative short-time existence was previously in-vestigated by Chan-Chen-Lee [4, Theorem 1.2]. For the sake of discussion,we’ll discuss the non-variable version here, which essentially assumes thefollowing hypotheses:uniform lower bound on Ricci curvature,uniform local bounds on the entropy, anduniform smallness of the local curvature concentration.(1.10)In [4] it is also assumed that the initial manifold has uniformly boundedcurvature. This raises two questions:1 Can the bounded curvature assumption be dropped?2 To what extent can the hypotheses (1.10) be weakened, while stillgiving short-time Ricci flow existence?Both questions were examined by Chan-Huang-Lee [5] who were ableto remove the bounded curvature assumption, and also weaken the lowerRicci bound assumption to a lower bound on scalar curvature and volumeupper bound estimates on a fixed scale. Weakening the Ricci curvaturebound was a simple observation about what properties were actually neces-sary. Addressing question 1 on the other hand is much more subtle. Theywere able to show this result by showing that the set of hypotheses (1.10)is semi-preserved by the process of “blowing up the boundary” (introducedby Hochard [25]) of a compact subset of a potentially unbounded curvatureinitial manifold, which allows one to obtain a sequence of approximatingbounded curvature metrics with (essentially) the same hypotheses. Thenshort-time Ricci flow existence for the unbounded curvature initial metriccan be obtained via Hamilton’s Compactness and the sequence of flows com-ing from the approximating sequence. This same argument is used in theproof of Theorem D, see Section 4.5.Despite the remarkable nature of the proof in [5] not requiring boundedcurvature, the authors of [5] do impose that the initial manifold should enjoy16an ineffective lower Ricci bound, which is used in the creation of a smoothdistance-like function. To summarize precisely, the initial metric g shouldsatisfy Rg ≥ −λ,VolgBg(x, r) ≤ v0rn for all x ∈M, r ∈ (0, 1],ν¯(Bg(x, 1), g, τ) ≥ −A for all x ∈M,(∫Bg(x,1)|Rm|n2g dVg) 2n ≤ σ for all x ∈M, andinfx∈MRcg(x) > −∞,(1.11)where λ, v0, τ, A can be prescribed arbitrarily, but σ depends on these choices.We direct the reader to Section 2.1 for the definition of the localized ν- andν¯-functionals. As mentioned above, the ineffective lower Ricci bound in(1.11) is expected not to be necessary as it is only ever used to constructa smooth distance-like function on M and, as commented on in [5], thisassumption may be weakened to a condition on how quickly Rc−g grows atspatial infinity.Rather than investigating the relationship between Ricci flow smoothingand the ineffective lower Ricci bound however, we instead further addressquestion 2 by showing that the scalar curvature bound assumption in (1.11)is redundant in obtaining quantitative short-time existence in the sense thatthe required smallness of σ and the estimates of the solution need not dependon λ. We prove the following.Theorem E. For all n ≥ 4, A, τ, v0 > 0, there are C0, σ, T > 0 dependingonly on n,A, τ, v0 such that the following holds. Let (Mn, g) be a completenoncompact manifold (not necessarily bounded curvature) such that for someϵ ≤ σ and all x ∈M , the metric g satisfies:(i) VolgBg(x, r) ≤ v0rn for all r ≤ 1;(ii) ν¯(Bg(x, 1), g, τ) ≥ −A;(iii)(∫Bg(x,1)|Rm|n2g dVg) 2n ≤ ϵ; and(iv) infx∈MRcg(x) > −∞.17Then there exists a complete Ricci flow g(t) on M × [0, T ] with g(0) = gsuch that for all (x, t) ∈M × (0, T ], there holds|Rm|g(t)(x) ≤ C0ϵt−1,injg(t)(x) ≥ C−10√t, and(∫Bg(0)(x,116) |Rm|n2g(t) dVg(t)) 2n ≤ C0ϵ.(1.12)Before we provide the applications of Theorem E, we first make someremarks regarding its hypotheses and statement.• We have already mentioned above that the ineffective lower Riccibound (iv) is likely a redundant hypothesis as it is only used to con-struct a distance-like function on (M, g) satisfying some desired prop-erties. Regarding the other assumptions, the C0-control in the formof a ν¯-functional bound (ii) and the proportional smallness of the cur-vature concentration (iii) are a somewhat natural pairing from a Ricciflow point of view (see the previous introductory sections as well asChapter 2 for more discussion on this). So that leaves the volumebound assumption (i) as the ‘odd-one-out’. I would like to suggestthat this is not a coincidence: It is my belief that assumption (i)should also be redundant (here and in Theorems F and G as well).For example, (i) can be removed if we had a version of our ExpandingBalls Lemma 5.1.3 that gave a continuous rate of expansion of balls attime zero (see Conjecture 5.1.4 for a precise statement that we coulduse). The proof of this claim is given below, see Claim 1.4.1. More-over, we discuss removing the volume assumption in the global settinglater on, see the discussion after Question 1.4.3.• The choice of r = 116 in the curvature concentration in (1.12) is forconvenience, not necessity; any r < 1 would work with appropriatemodification. Also, by applying the Shrinking Balls Lemma [41], thedomain in the integral could be replaced with the ball of radius r withrespect to g(t) after possibly shrinking T further.• The constants C0, σ, T in Theorem E can be explicitly bounded interms of n,A, τ, v0, though we do not pursue this here because ourmethods for bounding these variables would be far from sharp. Forexample, suppose τ = 1 and v0 = 2ωn, just for simplicity. Then, sincethe local curvature concentration ought to be preserved by the flow (up18to a dimensional constant perhaps), we can compute that an optimalbound on σ, from the parabolic point of view at least, should beσ ≥ 1C(n)e2An. (1.13)This expected bound on σ comes from the equivalence between the ν¯-functional and an L2-Sobolev inequality, and the relationship betweenthis Sobolev inequality and the curvature concentration via Ho¨lder’sinequality. The reason why we would be unable to achieve a boundof the form (1.13) with our current machinery is due to the fact thatthere is not a known sharp relationship between volume ratios and L2-Sobolev inequalities in spaces with possibly negative Ricci curvature.We direct the reader to Chapter 2 and Section 4.5 for more relateddiscussion.Before moving on to the applications of Theorem E, we first prove theclaim given in the first bullet point above regarding removing the uppervolume bound assumption.Claim 1.4.1. By possibly reducing σ, the upper volume bound assumption(i) in Theorem E can be made redundant in the presence of a version of theExpanding Balls Lemma with a continuous rate of expansion.Proof. Given any fixed A, τ > 0 and v > 1, assume for the sake of contradic-tion that there exists a sequence of complete pointed Riemannian manifolds(Mi, gi, xi) and σi → 0 which satisfyinfx∈M ν¯(Bgi(x, 1), gi, τ) ≥ −A,supx∈Mi∫Bgi (x,1)|Rm|n2gidVgi ≤ σi,VolgiBgi(xi, 1) ≥ vωn,infx∈Mi Rcgi > −∞.(1.14)Then by the assumed ineffective lower Ricci bounds, there is some maximal(and finite) ri > 0 (possibly different for each i ∈ N) such thatsupx∈MVolgiBgi(xi, r) ≤ vrn for all r ≤ ri.So by scaling up by a factor of r−2i we can replace our assumptions (1.14)19withinfx∈M ν¯(Bgi(x, 1), gi, τ) ≥ −A,supx∈Mi∫Bgi (x,1)|Rm|n2gidVgi ≤ σi,supx∈M VolgiBgi(x, r) ≤ vωnrn for all r ≤ 1 (equality if r = 1), andinfx∈Mi Rcgi > −∞.(1.15)At this point, we shift the base points xi (if necessary) so that we mayassume thatVolgiBgi(xi, r) ≥(1 + v2)ωnrn (1.16)Using our modified assumptions (1.15), we may apply our short-time exis-tence Theorem E to each (Mi, gi), followed up with Hamilton’s Compact-ness Theorem (coupled with a diagonal argument to work backwards to timezero), to conclude that the (Mi, gi, xi) subconverges in the Cheeger-Gromovsense to a limiting complete flow (M˜, g˜(t), x∞), t ∈ (0, T˜ ] that is necessarilyflat. Now take ϵ > 0 small enough so that(1 + ϵ)2+n <(1 + v2). (1.17)Then by assuming t > 0 is chosen adequately small, we may apply ourVolume Perturbation Lemma 5.1.1, the above Expanding Balls Conjecture5.1.4, the convergence gi(t) to a flat metric g˜(t), as well as (1.16) and (1.17),to obtainvωn = Volgi(0)Bgi(0)(x0, 1)≤ (1 + ϵ)Volgi(t)Bgi(0)(x0, 1)≤ (1 + ϵ)Volgi(t)Bgi(t)(x0, 1 + ϵ)≤ (1 + ϵ)2+nωn < vωn,which is absurd.We also give two applications of our short-time existence theorem. Thefirst of which can be viewed either (a) as a Ricci flow smoothing result formeasure spaces (M,µ) that are measure-space limits of a sequence of smoothmanifolds (Mi, gi) satisfying Theorem E, or (b) as a Gromov-Hausdorff com-pactness result for the family of metrics satisfying the hypotheses of TheoremE. One important thing to point out about Theorem F is that the only L∞20assumption on the curvature is that each gi satisfies some uniform lowerRicci bound, but that bound need not be the same for each i ∈ N.Theorem F. For any n ≥ 4, A, τ, v0 > 0, let (Mni , gi, pi) be any sequence ofcomplete pointed Riemannian manifolds (not necessarily bounded curvature)each satisfying hypotheses (i) - (iv) from Theorem E. Then there exists asmooth pointed manifold (M∞, x∞) endowed with a complete distance metricd0 (which generates the same topology as the smooth structure of M∞) anda Radon measure µ such that, up to a subsequence,(Mi, dgi ,Volgi , pi) −→ (M∞, d0, µ, x∞) (1.18)where the convergence is as pointed metric measure spaces (as describedprecisely below). Moreover, there exists a complete Ricci flow g(t) on M∞×(0, T (n,A, τ, v0)] satisfying the estimates (1.12) which emerges from (M∞, µ)in the weak sense, i.e., that for any precompact Borel set E ⊂ M∞, thereholdslimt→0Volg(t)(E) = µ(E).The notion in which the convergence (1.18) holds is as follows. There issome subsequence {ik}k∈N of N, an exhausting sequence Ωk ⊂ M∞ of opensets, and functions Fk : Ωk →Mik (each of which are diffeomorphisms ontotheir images) with Fk(x∞) = pik which satisfylimk→∞VolF ∗k gik (E) = µ(E)for any precompact Borel set E ⊂M∞, and(Mik , dgik , pik)pointed Gromov-Hausdorff−−−−−−−−−−−−−−−−→ (M∞, d0, x∞),where the Fk’s may serve as the approximating functions.Remark 1.4.2. Although the convergence of the distance function dg(t)back to the ‘time 0’ data d0 is not guaranteed by Theorem F, it is cer-tainly expected that this is also true. The shortfall here is due to the factour version of the Expanding Balls Lemma (Lemma 5.1.3) does not give acontinuous rate of expansion at the initial time. See also Question 2 below.21The second application we present is for manifolds (M, g) that satisfythe hypotheses of Theorem E on all scales, thus allowing us to apply thetheorem to increasingly extreme blow-downs to obtain long-time Ricci flowexistence with the scale invariant properties (1.12) holding for all times.This allows us to conclude topological rigidity and global volume bounds onthe initial manifold.Theorem G. For all n ≥ 4, A, v0 > 0, there exists σ > 0 depending onn,A, v0 such that the following holds. For any ϵ ≤ σ, we denote Mϵ tobe the family of complete noncompact (not necessarily bounded curvature)Riemannian manifolds (Mn, g) satisfying:(i) VolgBg(x, r) ≤ v0rn for all x ∈M, r > 0;(ii) ν¯(M, g) ≥ −A;(iii)(∫M |Rm|n2g dVg) 2n ≤ ϵ; and(iv) infx∈MRcg(x) > −∞.Then each of the following rigidity properties holds:(a) If (Mn, g) ∈Mσ, then Mn is homeomorphic to Rn, and if n ̸= 4 thenthe homeomorphism can be taken to be a diffeomorphism.(b) For any (M, g) ∈ Mσ, the volume growth of (M, g) at infinity is atleast that of Euclidean space, uniformly with respect to the base point.More precisely,lim infr→∞,x∈MVolgBg(x, r)rn≥ ωn.(c) By restricting ϵ smaller, we may assume that all volume ratios areuniformly close (from below) to that of Euclidean space, uniformlywith respect to the manifold, base point, and scale. More precisely,lim infϵ→0{VolgBg(x, r)rnwhere (M, g) ∈Mϵ, x ∈M, r > 0}≥ ωn.In addition to the obvious question of whether the ineffective lower Riccibound assumption (iv) can be removed from Theorem G, we also presenttwo additional questions.22Question 1.4.3. Is the upper bound on volume ratios assumption (i) alsoredundant in the statement of Theorem G?For example, if g is assumed to have bounded curvature, then conclusions(b) and (c) are still true even with assumption (i) removed, which followsfrom the statement and proof of Theorem A. Moreover, the aforementionedtheorem includes a topological rigidity statement stronger than conclusion(a), assuming an additional mild hypothesis on the curvature but withoutthe volume bound assumption. The primary issue that we face withoutthe bounded curvature assumption is that our method to show that thecurvature concentration remains bounded along the flow uses the ExpandingBalls Lemma 5.1.3, which seems to require an a priori upper bound onvolume ratios.Our other question is also related to an issue with the Expanding BallsLemma. Specifically, if we had a version of this lemma that gave a continuousrate of expansion akin to that of the Shrinking Balls Lemma (instead ofsimply the conclusion that Bg(0)(x, 1) ⊂ Bg(t)(x,R), see Section 5.1 for moredetail), then we would be able to prove that the analogous almost-Euclideanupper bound conclusions of (b) and (c) would also be true. In other words,we would have an affirmative answer to the following.Question 1.4.4. With the same notation ofMϵ from Theorem G, is it truethatlim supr→∞,x∈MVolgBg(x, r)rn≤ ωn,for any (M, g) ∈Mσ and thatlim supϵ→0{VolgBg(x, r)rnwhere (M, g) ∈Mϵ, x ∈M, r > 0}≤ ωn?We expect that both Question 1.4.3 and Question 1.4.4 can be answeredin the affirmative, but (to the author’s knowledge), their answers are bothcurrently unknown.1.5 Notation and conventionsThroughout this dissertation, we will assume intimate knowledge ofsmooth and Riemannian manifolds. Our particular choices of notation are23as follows. Given a n-dimensional Riemannian manifold (Mn, g), we willwriteRg, Rcg, Rmgfor the scalar, Ricci, and Riemannian curvature tensors respectively. Wecan relate the scalar curvature to the norm of the curvature tensor throughthe inequality|Rg| ≤ n|Rm|g, (1.19)which is a common inequality in Riemannian Geometry, see [27, Theorem7.28] for instance.Given a metric g, the symbols∇g, ∆gdenote its covariant derivative (applies to tensors) and Laplace-Beltrami(divergence of the gradient) operators respectively. We will write dVg forthe volume form and denoteVolg(Ω) =∫ΩdVg.The Riemannian g-distance function from a point x ∈ M is denoted bydg(x, ·), and the Riemannian r-ball centered at x is denoted by Bg(x, r).When it will not cause confusion, we may often suppress the metric gfrom notation, opting instead for the symbolsR, Rc, Rm, ∇, ∆, dV, Vol, etc.More on suppressing notation: Given a smooth function f(x, t) definedon Mn × [0, T ], we will often abbreviate as∫f dVt :=∫Mf(x, t) dVg(t)(x).For example, something like∫|∇u|2 +R+ |Rm|n2 dVtshould be interpreted as∫M|∇g(t)u|2g(t) +Rg(t) + |Rmg(t)|n2g(t) dVg(t).24Regarding common function spaces, we will use the following symbols:Ck = space of k times continuously differentiable functions;Lp = space of functions with∫|f |p <∞;W k,p = space of k times weakly differentiablewith each weak derivative in Lp (‘Sobolev space’);C∞ = space of smooth (infinitely differentiable) functions;C∞c = space of C∞ functions with compact support; andW k,p0 = closure of compactly C∞c under the Wk,p norm.The symbols ∥ · ∥Lp , ∥ · ∥Ck , and ∥ · ∥L∞ denote the standard Lp, Ck, andsupremum norms respectively. Moreover, all functions henceforth are realvalued.25Chapter 2Sobolev DataIn this chapter, we will investigate the relationship between three differ-ent notions of “Sobolev control” in manifolds. These three different notionsare:1. L2-Sobolev inequalities.2. Control on the (local) ν-functional.3. Lower bounds of volume ratios (for manifolds with lower Ricci curva-ture bounds).The layout and primary results of this chapter are as follows:• In Section 2.1, we recall the definition of Perelman’s localized µ- andν-functionals. These are useful to Ricci flow theory because they are(semi)-preserved along the flow, assuming mild hypotheses on the cur-vature bounds of the flow.• In Section 2.2, we show that control on the (localized) ν-functional isimplied by a bound on the (localized) L2-Sobolev inequality. This isa standard analytical result.• In Section 2.3 we show that one can obtain from a bound on the(local) ν-functional a “sharp” bound on the L2-Sobolev inequality.This observation for manifolds was originally due to Ye [49], and hassince been expanded upon by several authors.By “sharp” here, we mean that there is no information lost, besidesthe introduction of a dimensional constant, when going from an L2-Sobolev inequality to ν-functional, back to an L2-Sobolev inequality.• In Section 2.4, we put the previous pieces together - equivalence of L2-Sobolev inequalities and ν-functional control, as well as the (semi)preservation of the latter - to see that L2-Sobolev inequalities are(semi) preserved along Ricci flows, assuming mild control on the cur-vature.26• Finally, in Section 2.5, we will see that under the extra hypothesesthat a manifold has nonnegative Ricci curvature Rcg ≥ 0, the previoustwo notions of Sobolev control are also equivalent (up to dimensionalconstants) to lower bounds on volume ratios. This subtle observationis critical to the proof of Theorem C.Before jumping into the details, we introduce the following notation forthe sake of brevity, which we will use throughout.Definition 2.0.1. For a connected Ω ⊂ (M, g), we will writeSob(Ω) ≤ [A,B]to mean that the L2-Sobolev inequality(∫Ω|u| 2nn−2 dVg)n−2n≤ A(∫Ω|∇u|2 +Bu2 dVg)holds for all u ∈ W 1,2(Ω) with suppu ⊂⊂ Ω. We may also write Sob(Ω, g)to emphasize which metric is being used.2.1 Perelman’s localized µ- and ν-functionalsWe begin by recalling the definition of local entropy formulated by Wang[42] (originally introduced for compact manifolds by Perelman [36]): Let(Mn, g) be a complete Riemannian manifold of dimension n, Ω ⊂ M aconnected open subset with smooth boundary, and defineS (Ω) :={u∣∣∣∣u ∈W 1,20 (Ω), u ≥ 0, ∫Ωu2 dVg = 1},W (Ω, g, u, τ) :=∫Ω4τ |∇u|2 − u2 log u2 dVg − n2log(4πτ)− n,W (Ω, g, u, τ) :=∫Ωτ(4|∇u|2 +Ru2)− u2 log u2 dVg − n2log(4πτ)− n,µ¯(Ω, g, τ) := infu∈S (Ω)W (Ω, g, u, τ), µ(Ω, g, τ) := infu∈S (Ω)W (Ω, g, u, τ),ν¯(Ω, g, τ) := infs∈(0,τ ]µ¯(Ω, g, s), ν(Ω, g, τ) := infs∈(0,τ ]µ(Ω, g, s),ν¯(Ω, g) := infs∈(0,∞)µ¯(Ω, g, s), ν(Ω, g) := infs∈(0,∞)µ(Ω, g, s).27Although these definitions are given for domains Ω with smooth bound-aries, we will often take the domain to be some geodesic ball withinM , whichwill not necessarily have smooth boundary. This slight abuse of notation isvalidated for our purposes by [42, Lemma 2.6 and Remark 2.7].It is well known that the Ricci flow corresponds to a gradient flow ofthe µ-functional and therefore satisfies a monotonicity formula. This mono-tonicity was first shown to hold for compact flows by Perelman [36] andhas since received a lot of attention for the noncompact case (cf. [8], [50],[13], etc.). The following lemma summarizes this monotonicity property forcomplete bounded curvature Ricci flows.Lemma 2.1.1. Suppose that (Mn, g(t)), t ∈ [0, T ] is a complete boundedcurvature Ricci flow. Then for any t ∈ [0, T ] and τ > t, there holdsµ(M, g(0), τ) ≤ µ(M, g(t), τ − t).When Ω is not equal to all ofM however, the question of monotonicity (oralmost monotonicity) is much more subtle. It turns out that this “almostmonotonicity” is still true, and the proof involves careful choice of cutofffunction and estimates of the distance function under C/t curvature control.The following is [15, Lemma 2.5], which is a just a somewhat sharpenedversion of [42, Theorem 5.4].Lemma 2.1.2 ([15] Almost monotonicity of the local µ-functional). Let(M, g(t)), s1 ≤ t ≤ s2 be a complete bounded curvature Ricci flow satisfyingt · Rc(x, t) ≤ (n− 1)A, for all x ∈ Bg(t)(x0,√t).Then for any 0 ≤ s1 < s2 ≤ 1, τ1 > 0, 0 ≤ B < 12 , and 0 ≤ D < 8 − 20B,we haveν(Ω′′s2 , g(s2), τ1 + 1− s2)− ν (Ωs1 , g(s1), τ1 + 1− s1)≥ −(τ1 + 110A2B2+ e−1)·(es2−s110A2B2 − 1),whereΩ′′s2 = Bg(s2) (x0, 10A(1− 2B)− 2A√s2 −DA)andΩs1 = Bg(s1) (x0, 10A− 2A√s1 −DA) .28In fact, Lemma 2.1.2 is much more general than Lemma 2.1.1, which onecan see by taking B small (for any given time interval) and A → ∞ andapplying the lower-continuity of the local µ-functional [42, Proposition 2.3].Remark 2.1.3. Due to the normalization of∫Ω u2dV = 1, it is immediatefrom the definition of the µ¯-functional thatµ¯(Ω, g, τ) ≥ −Aimplies the log-Sobolev inequality∫Ωu2 log u2 dVg ≤∫Ω4τ |∇u|2 dVg + (A− n2log(4πτ)− n)=∫Ω4τ |∇u|2 dVg + (A− n2log(4πτ)− n)∫Ωu2 dVg.For this reason, we may often refer to control on these functionals as simplya “log-Sobolev inequality”.Remark 2.1.4. A common theme that we will see throughout this workis that the power of n/2 of the curvature |Rm| is precisely the one thatis necessary to “play nicely” with a scalar curvature weighted L2-Sobolevinequality (∫Ω|u| 2nn−2 dVg)n−2n≤ A(∫Ω|∇u|2 +Ru2 dVg)by means of Ho¨lder’s inequality. More precisely, by applying Ho¨lder’s in-equality (with exponents n2 ,nn−2), one obtains the inequality∣∣∣∣∫ΩRu2 dVg∣∣∣∣ ≤ (∫Ω|R|n2 dVg) 2n(∫Ωu2nn−2 dVg)n−2nThus if it is known that ∥R∥Ln2≤ 12A , then this term can be absorbed backinto the left-hand side of our Sobolev inequality to obtainSob(Ω) ≤ [2A, 0].In this sense, the specific smallness of the curvature concentration(∫Ω|R|n2 dVg) 2n≲(∫Ω|Rm|n2 dVg) 2n≲ A−1is the “correct” assumption to make, at least in light of Ho¨lder’s inequalityand the relationship between the ν-functional and L2-Sobolev inequalities(as will be described in the subsequent sections).292.2 ν-functional from Sobolev inequalityThe purpose of this section is to show that an L2-Sobolev inequalityimplies control on the ν¯-functional, or equivalently in our setting where thecurvature concentration is considered to be small (as pointed out in Remark2.1.4), control on the more classical ν-functional. The results of this sectionare simple consequences of Jensen’s inequality and are well known, see [49],[13] for instance.Lemma 2.2.1 (ν¯-functional from Sobolev). For any n ≥ 3, there exists adimensional constant C(n) > 0 such that the following holds. Let (Mn, g) bea (possibly incomplete) Riemannian manifold with Ω ⊂⊂ M and Sob(Ω) ≤[A, aτ ] for some A ≥ 1, a ≥ 0, τ > 0. Thenν¯(Ω, g, τ) ≥ −n2log(C(n)A)− 4a.If in addition∫Ω |Rm|n2 dVg ≤(2nA)n2 , or more generally∫Ω |R|n2 dVg ≤(2A)n2 , thenν(Ω, g, τ) ≥ −n2log(C(n)A)− 4a.Proof. Just as in the proof of [49, Theorem 3.3], Jensen’s inequality and thefact that log x ≤ bx− 1− log b (for any b > 0) implies∫Ωu2 log u2 dVg≤ n2log(∫Ω|∇u|2 + aτu2 dVg)+n2logA≤ 4s(∫Ω|∇u|2 + aτu2 dVg)+n2logA− 1− log(8sn).(2.1)This shows that ν¯(Ω, g, τ) ≥ −n2 log(C1(n)A) − 4a, because 4saτ ≤ 4a whens ≤ τ .For the other implication, we apply Ho¨lder’s and our L2-Sobolev inequal-ity to bound the scalar curvature term2s∫ΩRu2 dVg ≥ −4s(∫Ω|∇u|2 + aτu2 dVg).30Putting this back into (2.1) gives∫Ωu2 log u2 dVg ≤ 8s(∫Ω|∇u|2 + R4u2 +aτu2 dVg)+n2logA− 1− log(8sn).Now relabel s˜ = 2s and restrict s˜ ≤ τ to obtainν(Ω, g, τ) ≥ −n2log(2C1(n)A)− 4a.For simplicity, we will assume C(n) = 2C1(n) from which both implicationsclearly follow.As a corollary to the statement and proof of Lemma 2.2.1, we also statethe following version of that result assuming that the manifold is completeand that a = 0. We choose to take the time to state it in the precise formbelow due to the ease at which it may be applied in the proof of TheoremA later.Corollary 2.2.2. Let (Mn, g), n ≥ 3 be any complete bounded curvatureRiemannian manifold satisfyingSob(M, g) ≤ [Cg, 0].Then each of the following is true:(1) For all compactly supported u ∈ W 1,2(M, g) with ∫ u2 dVg = 1 and anyτ > 0 there holds∫u2 log u2 dVg ≤ 4τ∫|∇u|2 dVg − n2log τ +n2(logCg + logn8− 1).(2) Assume moreover that(∫|R|n2 dVg) 2n≤ 2Cg.Then for all compactly supported u ∈W 1,2(M, g) with ∫ u2 dVg = 1 and anyτ > 0 there holds∫u2 log u2 dVg ≤ 4τ(∫|∇u|2 + R4u2 dVg)− n2log τ +n2(logCg + logn4− 1).312.3 Sobolev inequality from ν-functionalThe purpose of this section is to show that given a log-Sobolev inequal-ity, one can obtain an L2-Sobolev one in return. Moreover, the preciserelationship between the relevant constants are “sharp” in the sense thatno information is lost (besides introducing a dimensional constant) if onegoes from L2-Sobolev to log-Sobolev, then back to an L2-Sobolev inequal-ity. This is particularly useful when it comes to Ricci flow theory becausethe ν-functional is the quantity that is preserved under the flow (recall Lem-mas 2.1.1 and 2.1.2) while L2-Sobolev inequalities are what are most usefulfrom a computational/analytical point-of-view.The method of obtaining an L2-Sobolev inequality from a log-Sobolevinequality is well known in the literature: Davies [19, Chapters 1-2] describesthe process in the general context of abstract measure spaces (Ω,dx); Ye [49,Theorems 5.3-5.5] works out this process for compact Riemannian manifolds(M, g), showing in particular that the Sobolev constant obtained does notdepend on the geometry of g; and Chen [14, Appendix B] extends this toasymptotically flat manifolds (M, g) by applying Ye’s result to obtain auniform Sobolev inequality for each member of an exhaustion by compactsets Ω1 ⊂ Ω2 ⊂ · · · ⊂M , then extending this to a Sobolev inequality on Mby this uniformity in j, and the fact that W 1,20 (M, g) =⋃j∈NW 1,20 (Ωj , g).We observe that in this argument, the condition of asymptotic flatness(as was used in [14]) can actually be weakened to some connected componentΩ of a possibly incomplete Riemannian manifold M . In summary, we havethe following.Lemma 2.3.1 (Sobolev from ν¯-functional). For any n ≥ 3, there existsa dimensional constant C(n) > 0 such that the following holds. Suppose(Mn, g), is a (possibly incomplete) Riemannian manifold, Ω ⊂⊂ M is con-nected, and A, τ > 0.1. If ν¯(Ω, g, τ) ≥ −n2 logA, then Sob(Ω) ≤ [C(n)A, 1τ ].2. If ν(Ω, g, τ) ≥ −n2 logA and∫Ω |Rm|n2 dVg ≤(2nC(n)A)n2, or moregenerally∫ΩRn2 dVg ≤(1C(n)A)n2, then Sob(Ω) ≤ [2C(n)A, 1τ ].Proof. We will only discuss the proof of part (2) since part (1) is simpler.Fix some λ > 0, a compact K ⊂⊂ Ω, and observe that the condition on the32ν-functional implies∫Ωu2 log u2 dVg≤ s(∫Ω|∇u|2 + R4u2 dVg)− n2log s+n2logA− n2log(π)− n< s(∫Ω|∇u|2 + R+4u2 + λu2 dVg)− n2log s+n2logA− n2log(π)− nfor all s ∈ (0, 4τ ] and u ∈ W 1,20 (K) with∫K u2 dVg = 1. Now by [49,Theorem 5.5], we may conclude an L2-Sobolev inequality(∫Ku2nn−2 dVg)n−2n≤ C(n,A)(∫K|∇u|2 + R+4u2 +1τu2 + λu2 dVg) (2.2)for all u ∈ W 1,20 (K). Even though the results in [49] are stated for closed(compact without boundary) manifolds, equivalent statements hold for com-pact manifolds with boundary, such as our K ⊂⊂ Ω (see, for instance, [14,Appendix B]). Now since K is compact and C(n,A) has no dependence onλ, we may send λ→ 0 (by, say, the Dominated Convergence Theorem).The fact that the constant C(n,A) is of the precise form C(n)A followsby tracking through the constants in [49, Theorems 5.3-5.5] as was firstnoticed by Chen [13] (see also Section 3.4). So it remains to deal with theremaining scalar curvature term. For this, we apply Ho¨lder’s inequality:C(n)A4∫KR+u2 dVg ≤ C(n)A4(∫K|R|n2 dVg) 2n(∫Ku2nn−2 dVg)n−2n≤ 12(∫Ku2nn−2 dVg)n−2n.So we can absorb this term into the left-hand side of (2.2) to giveSob(K) ≤ [2C(n)A, 1τ].The result follows by taking an exhaustion of compact K1 ⊂ K2 ⊂ · · · ⊂ Ωbecause the right-hand side above does not depend on K.332.4 Sobolev inequality along the flowThe purpose of this section is to show that L2-Sobolev inequalities aresemi-preserved along the Ricci flow, assuming that the curvature concentra-tion is assumed to remain uniformly small (proportional to the inverse ofthe L2-Sobolev constant) along the flow.As alluded to in the previous sections, this preservation is achieved byfirst translating the L2-Sobolev inequality to a log-Sobolev inequality, whichin turn gives control on the (local) ν-functional (by the initial smallness of∥R∥Ln2) - see Section 2.2. This quantity is then (almost) monotone alongthe flow - see Section 2.1. Then using the assumed smallness of ∥R∥Ln2at alater time t > 0, we can transport back to an L2-Sobolev inequality at timet - See Section 2.3.So in essence, there is nothing new to be said here. Instead, we will statea theorem summarizing these results in a clean way for complete boundedcurvature flows. The following was first written down for compact manifoldsby Ye [49] and subsequently for asymptotically flat manifolds by Chen [13].As noted in the proof of Lemma 2.3.1 however, the proof generalizes effort-lessly to noncompact complete bounded curvature flows. We will provide abrief proof for the sake of completeness and for ease of reference in the proofof Theorem A.Theorem 2.4.1. For n ≥ 3, there exists a dimensional constant L(n) >0 such that the following holds. Suppose that (Mn, g(t)), t ∈ [0, T ] is acomplete bounded curvature Ricci flow and that the initial metric g(0) =g0 satisfies the Sobolev inequality Sob(M, g0) ≤ [Cg0 , 0] and the curvaturecondition (∫M|R|n2 dVg0) 2n≤ 2Cg0.Then with A = L(n)Cg0, the weighted Sobolev inequality(∫|u| 2nn−2 dVg(t))n−2n≤ A(∫|∇u|2 +R+u2 dVg(t))(2.3)holds for all compactly supported u ∈W 1,2(Mn, g(t)) on [0, T ].Proof. Part (2) of Corollary 2.2.2 gives us the following log-Sobolev inequal-34ity for the metric g0:∫u2 log u2 dVg0 ≤ 4τ(∫|∇u|2 + R4u2 dVg0)− n2log τ +n2(logCg0 + logn4− 1).Monotonicity of the µ-functional (Lemma 2.1.1) implies that precisely thesame log-Sobolev inequality holds as well for any t ∈ [0, T ] and any τ > 0.Now given a t ∈ [0, T ], we replace Rg(t) with R+g(t) to obtain∫u2 log u2 dVg(t) ≤ 4τ(∫|∇u|2 + R+4u2 dVg(t))− n2log τ +n2(logCg0 + logn4− 1).(2.4)We are then in a position to apply Proposition 2.3.1 - or, more accurately,its proof (see (2.2) in particular) - which allows us to conclude that (2.3) istrue at any given time t. Since the right-hand side of (2.4) is independent oft ∈ [0, T ], this establishes the result. As mentioned in the proof of Lemma2.3.1, the fact that A depends linearly on Cg0 is realized by tracking theconstants in [49] which is a computation that we carry out in Section 3.4wherein we also provide a bound on L(n).Unfortunately, we will not always be operating under the assumption ofa global Sobolev inequality and globally small curvature concentration, butrather we will often have to make do with local bounds of these quantities- see Chapters 4 and 5. In this case, the local almost-monotonicity Lemma2.1.2 has to replace the global monotonicity Lemma 2.1.1 in the synonymouslocal-L2-Sobolev preservation as the above lemma. We won’t give a precisestatement of this local preservation here because it will be easier to dealwith on a case-by-case basis. We will, however, round out this section withthe following remark about this process.Remark 2.4.2. Suppose g(t), t ∈ [0, T ] is a Ricci flow with an initialSobolov inequality Sob(Ω, g(0)) ≤ [A, 0] and some assumed smallness of∫ |Rm|n2 dVg(t) along the flow. Then successive application of Lemma 2.2.1and Lemma 2.3.1 will return us to an L2-Sobolev inequality for g(t):Sob(Ω, g(t)) ≤ [2C2(n)C1(n)A, 1τ].35It follows from our work in Sections 3.4 and 3.4.1 (in particular, the boundon L(n) in (3.34)) that C1(n)C2(n) ≤ 500n when n ≥ 4. Here C1 and C2are the constants from Lemma 2.2.1 and 2.3.1 respectively. This estimatewill be used to obtain the explicit bound for ϵ(n) in Theorem C.2.5 Sobolev inequality in nonnegative RiccicurvatureIn this section, we will show that under the additional hypothesis that(M, g) has nonnegative Ricci curvature Rcg ≥ 0, the equivalence betweenL2-Sobolev inequalities and ν-functional control (as described in previoussections) can, remarkably, be extended to include the supposedly weakerassumption of lower bounds on volume ratios as well, as in Figure 2.1.Figure 2.1: Nonnegative Ricci completes the circle of equivalencesThe fact that an L2-Sobolev inequality implies control on volume ratiosis well known, but since we care about the exact relationship between thesequantities, we will provide a short proof. The following is only a slightmodification of [49, Theorem 6.1].Lemma 2.5.1 (Volume from Sobolev). Let (Mn, g), n ≥ 3 be a (possiblyincomplete) Riemannian manifold with Bg(x, L) ⊂⊂ M for some L > 0.Assume that Sob(Bg(x, L)) ≤ [A,B] for some A > 0, B ≥ 0. Then wheneverr ≤ L, there holdsVolgBg(x, r) ≥(1A(2n+3 + 2BL2))n2rn.36Proof. Assume for the sake of contradiction that there exists (Mn, g, x),L > 0, A > 0, B ≥ 0 for which the hypotheses hold, and some r ∈ (0, L]such thatVolgBg(x, r) <(1A(2n+3 + 2BL2))n2rn =: δrn.Set g˜ = r−2g so that Volg˜Bg˜(x, 1) < δ. Then by the assumed L2-Sobolevinequality for the original metric g and the fact that r ≤ L, we obtainSob(Bg˜(x, 1)) ≤ [A,BL2]. Now by Ho¨lder’s inequality, Volg˜Bg˜(x, 1) < δ,and the definition of δ, we haveABL2∫Bg˜(x,1)u2 dVg˜ ≤ ABL2δ 2n(∫Bg˜(x,1)|u| 2nn−2 dVg˜)n−2n≤ 12(∫Bg˜(x,1)|u| 2nn−2 dVg˜)n−2n.Therefore this term can be absorbed into the left-hand side of the L2-Sobolevinequality for g˜ to give Sob(Bg˜(x, 1)) ≤ [2A, 0]. The rest of the proof thenis verbatim to the latter half of the proof of [49, Theorem 6.1].In the presence of a (possibly negative) lower Ricci curvature boundRcg ≥ −K, Wang [42, Theorem 3.6] proved that one can obtain local controlon the ν¯-functional depending on the volume ratio:ν¯(Bg(x, r), g, r2) ≥ −10n2(1 +Kr) + log(VolgBg(x, r)ωnrn)n+1. (2.5)Here ωn is the volume of an n-dimensional Euclidean unit ball. In partic-ular, when Rcg ≥ 0, we can obtain global control on the ν¯-functional (andsubsequently a global Sobolev inequality) if we know that all the volumeratios of (M, g) are uniformly bounded from below. Such a manifold is saidto have positive asymptotic volume ratio, which we will denote byAVR(g) := limr→∞VolgBg(x0, r)ωnrn.It is well known that when Rcg ≥ 0, AVR(g) does not depend on the choiceof fixed point x0, is a well-defined number in [0, 1], and that AVR(g) = 1if and only if (Mn, g) is isometric to (Rn, gE) where gE is the Euclideanmetric.37Now suppose that (M, g) were some nonnegative Ricci curvature mani-fold with a known volume boundVolgBg(x, 1) ≥ v > 0.Then by (2.5) we may obtain ν¯(Bg(x, 1), g, 1) ≥ −C + log(v n+1n ), which inturn gives Sob(Bg(x, 1)) ≤ [C ′v−2(n+1)n , 1] by Lemma 2.3.1. Now if we returnto a bound on the volume via Lemma 2.5.1, we obtainVolgBg(x, 1) ≥ cvn+1,which is a much worse volume estimate than we started with as v → 0.Therefore one step along this process is not sharp with respect to v. Itturns out that this bluntness occurs in going from volume to ν¯-functionalin (2.5), and is rectified in the following lemma. Put another way, Lemma2.5.2 completes the circle of equivalences between volume ratios, Sobolevinequality, and control on the ν¯-functional (up to dimensional constants).Lemma 2.5.2 (Sobolev from volume). For any n ≥ 3, there exists a di-mensional constant C(n) > 0 such that the following holds. Let (Mn, g) be acomplete Riemannian manifold (not assumed bounded curvature) such thatRcg ≥ 0 and AVR(g) > 0. If, for some r > 0 and v > 0, there holdsVolgBg(x, r)rn= v, (2.6)then Sob(Bg(x, r)) ≤ [C(n)v− 2n , 0].Remark 2.5.3. The constant v in the volume ratio hypothesis (2.6) ispermitted to be much larger than the value of ωn · AVR(g). In fact, theassumption that AVR(g) > 0 in Lemma 2.5.2 is likely superfluous becausethe proof of [2, Theorem 1.1] (which we will be modifying) is local in natureuntil the end when they let r →∞ (instead we will take a determined valueof r). Regardless, the stated version will be sufficient for our purposes.Proof of Lemma 2.5.2. First note that by volume comparison, we havev2n=VolgBg(x, r)(2r)n≤ VolgBg(x, 2r)(2r)n≤ VolgBg(x, r)rn= v.38Therefore, it suffices to prove the lemma with v replaced byv0 :=VolgBg(x, 2r)(2r)n,at the expense of replacing the dimensional constant C(n) with 2nC(n).Fix some smooth nonnegative function u ∈ C∞(Bg(x, r)) with u ≡ 0 on∂Bg(x, r). Since the hypotheses and conclusion of the lemma are invariantunder scaling, we may assume that∫Bg(x,r)|∇u| dVg = n∫Bg(x,r)unn−1 dVg. (2.7)If this seems like a somewhat mysterious condition, we refer the reader tothe proof of [2, Theorem 1.1], which we will be modifying (note that thecoefficient of n in (2.7) is only there to match the convention in [2]). Westart with a couple elementary claims.Claim 2.5.4. VolgBg(x, r) ≤ (2r)n(1− (17)n)v0Proof of Claim. Let γ : [0,∞) → M be a geodesic ray with γ(0) = x, theexistence of which is guaranteed by the completeness of (M, g). Settingy = γ(3r/2), we therefore haveBg(y, r/2) ⊂ Bg(x, 2r) \Bg(x, r).See Figure 2.2 for this visualization.Figure 2.2: Fitting in a small ball along a geodesic ray39Then by volume comparison and the fact that Bg(y,72r) ⊃ Bg(x, 2r), we cansee thatVolgBg(y, r/2)(r/2)n≥ VolgBg(y, 7r/2)(7r/2)n≥ VolgBg(x, 2r)(7r/2)n≥ VolgBg(x, 2r)(2r)n(4/7)n= v0(4/7)n.We therefore obtain the upper boundVolgBg(x, r) ≤ VolgBg(x, 2r)−VolgBg(y, r/2)≤ (2r)nv0 − (2r/7)nv0= (2r)n(1− (1/7)n)v0.This completes the proof of the claim.Claim 2.5.5. There exists dimensional constants δ(n) > 0, C1(n) ≥ 1 suchthat for any measurable nonnegative function f : Bg(x, r)→ R, there holds1(2r)n∫Bg(x,r)(1 + f(x))n dVg ≤ (1− δ)v0 + C1(2r)n∫Bg(x,r)f(x)n dVg.Proof of Claim. Applying the Binomial Theorem gives∫Bg(x,r)(1 + f(x))n dVg ≤ VolgBg(x, r) +n∑k=1(nk)∫Bg(x,r)f(x)k dVg.For the 1 ≤ k ≤ n− 1 terms, we write pk = nk , qk = nn−k and apply Ho¨lder’sthen Young’s inequalities∫Bg(x,r)f(x)k dVg ≤(∫Bg(x,r)f(x)n dVg) 1pk(VolgBg(x, r))1qk≤ ϵqkkqkVolgBg(x, r) +1ϵpkk pk∫Bg(x,r)f(x)n dVg.We may choose the ϵk small enough so that40(1− (1/7)n)(1 +n−1∑k=1(nk)ϵqkkqk)≤ 1− 12(1/7)n.Therefore, by applying Claim 2.5.4, we can see that with δ(n) = 12(1/7)nandC1(n) = 1 +n−1∑k=1(nk)1ϵpkk pk,the claim holds.Now we recall some of the details from [2]. The following is taken outof the proof of their main theorem (specifically right up until the end oftheir Corollary 2.6 and the subsequent few lines), which we state as anindependent result for clarity.Proposition 2.5.6 (Proof of [2, Theorem 1.1]). Suppose that (M, g) satisfiesRcg ≥ 0 and AVR(g) > 0. Let D ⊂ M be some compact connected regionwith smooth boundary and u ∈ C∞(D) is nonnegative with u ≡ 0 on ∂Dand ∫D|∇u|dVg = n∫Dunn−1 dVg.Then for all ρ > 0, there holdsVolg{y ∈M : dg(x, y) < ρ for all x ∈ D} ≤∫D(1 + ρ u1n−1)ndVg.We apply Proposition 2.5.6 with2 D = Bg(x, r) and ρ = 3r to obtainVolgBg(x, 2r)= Volg{y ∈M : dg(p, y) < 3r for all p ∈ Bg(x, r)}≤∫Bg(x,r)(1 + 3r u1n−1)ndVg.Now divide both sides by (2r)n and apply Claim 2.5.5 with f = 3ru1n−1 toobtainv0 ≤ (1− δ(n))v0 + C1(2r)n∫Bg(x,r)(3r u1n−1)ndVg.2The smooth boundary condition can be accomplished by padding Bg(x, r) slightly bytaking Bg(x, r) ⊂ Dϵ, where Dϵ has smooth boundary and is contained in the ϵ-ball ofBg(x, r) (see [42, Lemma 2.6]), then take ϵ→ 0.41Rearranging and writing C2(n) =C13nδ2n givesv0 ≤ C2∫Bg(x,r)unn−1 dVg.Therefore, by applying our scaling condition (2.7), we immediately see∫Bg(x,r)|∇u|dVg = n∫Bg(x,r)unn−1 dVg≥ nv1n0 C− 1n2(∫Bg(x,r)unn−1 dVg)n−1n.(2.8)Thus we obtain an L1-Sobolev inequality for the given function u, whichis scale-invariant. So we may scale back to our original value of r andtherefore may assume that (2.8) holds for all nonnegative u ∈ C∞(Bg(x, r))with u ≡ 0 on ∂Bg(x, r).Now it is well known that an L1-Sobolev inequality implies an L2-Sobolevinequality (cf. [30]). We provide a brief proof for the reader’s convenienceand to verify the proper dependence on v0. By the L1-Sobolev inequality(2.8) and Cauchy-Schwarz, we have(∫Bg(x,r)u2nn−2 dVg) 2(n−1)n=(∫Bg(x,r)(u2(n−1)n−2) nn−1dVg) 2(n−1)n≤(nv1n0 C− 1n2)−2(∫Bg(x,r)|∇u 2(n−1)n−2 | dVg)2=((nv1n0 C− 1n2)−1 2(n− 1)n− 2)2(∫Bg(x,r)unn−2 |∇u|dVg)2≤ C(n)v−2n0(∫Bg(x,r)u2nn−2 dVg)(∫Bg(x,r)|∇u|2 dVg).We obtain the desired L2-Sobolev inequality by rearranging the above andapproximation. This completes the proof of the lemma.42Remark 2.5.7. By taking the ϵk = (2 · 7nnn+1)−1 in the proof of Claim2.5.5, we can calculate a naive bound on the dimensional constant in Lemma2.5.2 when n ≥ 4 ofC(n) ≤ 2n(2(n− 1)n(n− 2))2︸ ︷︷ ︸≤2C2n2≤ 2n+1(32)n2 · 7n︸ ︷︷ ︸=1/δnn+1(2 · 7nnn+1)n︸ ︷︷ ︸≥C12n≤ (10n)3n.This computation is critical in the explicit estimation of ϵ(n) in Theorem C.As previously mentioned, when Rcg ≥ 0, a positive lower bound onthe asymptotic volume ratio AVR(g) implies a lower bound on all volumeratios by volume comparison. Therefore Lemma 2.5.2 immediately yieldsthe following corollary.Corollary 2.5.8. For any n ≥ 3 and C(n) as in Lemma 2.5.2, the followingholds. Let (Mn, g), n ≥ 3 be a complete Riemannian manifold (not assumedbounded curvature) such that Rcg ≥ 0 and AVR(g) > 0. Then Sob(M, g) ≤[C(n)(ωnAVR(g))− 2n , 0].43Chapter 3Long-Time ExistenceThe primary purpose of this chapter is to prove Theorem A, which giveslong-time Ricci flow existence and topological rigidity in the presence ofglobal Sobolev bounds and globally small curvature concentration. Thisresult is instrumental in some of the results we discuss later. First, we recallthe statement of the theorem.Theorem. For n ≥ 3, there are dimensional constants δ(n) > 0 andC0(n) = (5e)n2 such that the following holds. Let (Mn, g) be a completebounded curvature manifold which satisfies the Sobolev inequalitySob(M, g) ≤ [Cg, 0],and the scale-invariant curvature pinching(∫M|Rm|n2 dVg) 2n≤ δ(n) 1Cg.Then the complete bounded curvature Ricci flow g(t) with initial conditiong(0) = g exists for all times t ∈ [0,∞) and satisfies the following curvatureestimates:(a) For all t ∈ (0,∞), there holdssupx∈M|Rm|(x, t) ≤ C0(n)t.(b) The singularity at infinity satisfieslimt→∞ t supx∈M|Rm|(x, t) = 0.Moreover, if |Rm|g ∈ Lp for some p < n2 , then g(t) converges locally smoothlyto a complete flat limit metric g(∞) on Mn with Euclidean volume growth,and thus Mn is diffeomorphic to Rn.44The outline of this chapter is as follows.• In Section 3.1, we show that the global curvature concentration∫M|Rm|p dVg(t)is an upper-continuous quantity along complete bounded curvatureRicci flows, which follows from a maximum principle argument.• In Section 3.2, we prove Theorem A by showing the long-time existenceof the Ricci flow, as well as the C/t and faster than 1/t curvature decayof the solution. The statement of the convergence to a limiting metricg(∞) that is isometric to the Euclidean flat metric follows by arguingthat the curvature is actually integrable to time infinity. This also givethe topological rigidity that M is diffeomorphic to Rn.• In Section 3.3, we prove Theorem B, which follows as a corollary toTheorem A via Hamilton’s Compactness.• In Section 3.4, we calculate an explicit bound on L(n), and thus alsoon δ(n). A key tool in the calculation is a sharpened version of theMarcinkiewicz Interpolation Theorem.• Finally, in Section 3.5, we justify Remark 1.2.4 with a constructionwherein we glue in countably many “wrinkled balls” into Euclideanspace in such a way that the curvature of the resulting manifold isunbounded, while the curvature concentration and Sobolev constantsare as arbitrarily close to those of Euclidean space as desired.3.1 Upper continuity of the Lp-curvatureIn this section, we will show that the curvature quantity∫M|Rm|p dVg(t)is upper-continuous along any complete bounded curvature Ricci flow, as-suming it is initially finite. This upper-continuity becomes important in thesubsequent section wherein we actually show that this quantity (for severaldifferent values of p) is nonincreasing along the flow, assuming it is small45enough initially compared to the initial Sobolev constant (see Lemmas 3.2.2and 3.2.4).For the rest of this section we fix the notation that (M, g(t)), t ∈ [0, T ]is a complete bounded curvature Ricci flow and we writeK = supM×[0,T ]|Rm|(x, t) <∞.Before beginning the proofs, we would like to point out that if we wereto restrict to p ≥ 2, then the results in this section would be much simpler.When 1 ≤ p < 2 however (for example if n = 3 and p = n2 ), there is anadded difficulty due to the fact that∂t|Rm|p = p2|Rm|p−2∂t|Rm|2,and the curvature is certainly permitted to be zero in places.We get around this as follows. First, choose a smooth positive functionv0 : M → R that is integrable (with respect to g0) and satisfies 0 < v0 ≤ 1.Now for any ϵ > 0, define vϵ to be the solution to the initial value problem{∂tvϵ = ∆g(t)vϵvϵ(0, x) = ϵv0(x).Coupling the well-known evolution∂t|Rm|2 ≤ ∆|Rm|2 − 2|∇Rm|2 + 16|Rm|3with the evolution of v2ϵ which satisfies∂tv2ϵ = ∆v2ϵ − 2|∇vϵ|2,we can write∂t(|Rm|2 + v2ϵ ) ≤ ∆(|Rm|2 + v2ϵ )− 2|∇Rm|2 − 2|∇vϵ|2 + 16|Rm|3.By Cauchy-Schwarz and Young’s inequality, there holds|∇(|Rm|2 + v2ϵ )|2= 4|Rm|2|∇|Rm||2 + 4v2ϵ |∇vϵ|2 + 8|Rm|vϵ⟨∇|Rm|,∇vϵ⟩≤ 4|Rm|2|∇|Rm||2 + 4v2ϵ |∇vϵ|2 + 8|Rm|vϵ|∇|Rm|| |∇vϵ|≤ 4|Rm|2|∇|Rm||2 + 4v2ϵ |∇vϵ|2 + 4|Rm|2|∇vϵ|2 + v2ϵ |∇|Rm||2= 4(|Rm|2 + v2ϵ )(|∇|Rm||2 + |∇vϵ|2).46So if we set K2ϵ = |Rm|2 + v2ϵ , we can see that4K2ϵ |∇Kϵ|2 ≤ 4K2ϵ (|∇|Rm||2 + |∇vϵ|2).Subsequently, the Kato inequality |∇|T || ≤ |∇T | gives∂tK2ϵ ≤ ∆K2ϵ − 2|∇Kϵ|2 + 16K3ϵ .We now use this evolution of K2ϵ to determine a useful evolution inequalityfor Kpϵ .Lemma 3.1.1. For any p ≥ 1, there holds∂tKpϵ ≤ ∆Kpϵ + 8pKp+1ϵ . (3.1)Proof. Denote α = p2 . Since Kϵ is a strictly positive function on M × [0, T ],we freely differentiate and see that∂tK2αϵ = ∂t(K2ϵ )α = αK2(α−1)ϵ ∂tK2ϵ≤ αK2(α−1)ϵ (∆K2ϵ − 2|∇Kϵ|2 + 16K3ϵ )= αK2(α−1)ϵ ∆K2ϵ − 2αK2(α−1)ϵ |∇Kϵ|2 + 16αK2α+1ϵ= ∆K2αϵ − 4α(α− 1)K2(α−1)ϵ |∇Kϵ|2 − 2αK2(α−1)ϵ |∇Kϵ|2 + 16αK2α+1ϵ≤ ∆K2αϵ + 16αK2α+1ϵ ,where in the last line we have used the fact that α ≥ 12 implies that −4α(α−1)− 2α ≤ 0.We can use the heat-type evolution of Kpϵ as in (3.1), along with themaximum principle for complete bounded curvature Ricci flows to obtainthe following.Proposition 3.1.2. Let p ≥ 1 and suppose that |Rm|g(0) ∈ Lpg(0). Then∫M|Rm|p dVt ≤ e(8p+n)Kt∫M|Rm|p dV0.Proof. First let Ω0 ⊂⊂ Ω1 ⊂⊂ · · · ⊂⊂ M be an exhausting sequence ofrelatively compact connected domains with smooth boundaries in M , andfor each j ∈ N let ϕj : M → [0, 1] be a smooth cutoff function which issupported in Ωj and ϕj ≡ 1 in Ωj−1. Now for any ϵ > 0 and for each j ∈ N,47we denote by uj : M × [0, T ]→ R the solution to the initial value Dirichletboundary condition problem∂tuj = ∆g(t)ujuj(x, 0) = ϕjKpϵ (x, 0)uj(x, t) = 0 if x ∈ ∂Ωj .Then by the maximum principle, we know that the uj are all nonnegativefunctions which are uniformly boundedsupM×[0,T ]uj(x, t) ≤ supx∈MKpϵ (x, 0) ≤ supx∈M(|Rm|2(x, 0) + ϵ2) p2 . (3.2)Now the Dominated Convergence Theorem and the evolution ∂tdVt = −RtdVttogether yieldddt∫Ωjuj dVt =∫Ωj∂tuj −Ruj dVt=∫Ωj∆uj −Ruj dVt≤∫Ωj∆uj + nK∫Ωjuj dVt.(3.3)Here we have applied the bound |R| ≤ n|Rm|, see (1.19). Now Green’sTheorem says that ∫Ωj∆uj dVt =∫∂Ωj∇v(t)uj dVtwhere v(t) is the outward unit normal on ∂Ωj relative to g(t). Then bythe nonnegativity of uj and the Dirichlet condition uj = 0 on ∂Ωj , thisboundary integral is nonpositive. Subsequently, putting this back into (3.3)shows that∫Muj dVt ≤ enKt∫Muj(x, 0) dV0 ≤ enKt∫M(|Rm|2 + ϵ2v20)p2 dV0. (3.4)For each fixed point in spacetime (x, t) ∈M×[0, T ], the sequence uj(x, t)is monotone increasing (another consequence of the maximum principle),and thus has a pointwise limit which we denote by u(x, t). By the uni-form boundedness (3.2), Schauder estimates, the Arzela`-Ascoli Theorem,48and a diagonal subsequence argument, we obtain that the limit function uis smooth on M × [0, T ) and solves the initial value problem{∂tu = ∆g(t)uu(x, 0) = Kpϵ (x, 0).Moreover, the integral bound (3.4) descends to u as well via the MonotoneConvergence Theorem. That is to say∫MudVt ≤ enKt∫(|Rm|2 + ϵ2v20)p2 dV0. (3.5)We also define u˜(x, t) = e8pt(K2+ϵ2)12 u(x, t) so that∂tu˜ = 8p(K2 + ϵ2)12 u˜+ e8pt(K2+ϵ2)12 ∂tu= 8p(K2 + ϵ2)12 u˜+ e8pt(K2+ϵ2)12∆g(t)u= 8p(K2 + ϵ2)12 u˜+∆g(t)u˜,and u˜(x, 0) = Kpϵ (x, 0). Comparing this with the evolution inequality (3.1),another application of the maximum principle shows thatKpϵ (x, t) ≤ u˜(x, t)for all (x, t) ∈M × [0, T ]. Integrating this and applying our integral bound(3.5) yields∫M|Rm|p dVt ≤∫MKpϵ dVt ≤∫Mu˜(x, t) dVt(x)= e8pt(K2+ϵ2)12∫Mu(x, t) dVt(x)≤ e(8p+n)t(K2+ϵ2)12∫M(|Rm|2 + ϵ2v20)p2 dV0.Now because of the assumed integrability of |Rm|pg(0) and v0 (with respectto g0), we can take ϵ→ 0 and once again apply the Dominated ConvergenceTheorem to obtain the result.493.2 Long-time existence and convergenceIn this section, we give the prove of Theorem A. The majority of theresults in this section follow the same outline as done by Chen [13, Section4] (though there is significant divergence near the end) and the proofs are(naturally) very similar or, in some cases even, are identical. For the read-ers’ convenience, we will present the proofs in full when there are notabledifferences, and will omit them otherwise.We now start the proof of Theorem A, and let (Mn, g) satisfy the hy-potheses of the theorem for some constant Cg ∈ (0,∞]. We also let g(t) onM × [0, Tmax) be the unique complete bounded curvature Ricci flow withg(0) = g, with existence guaranteed from [39] and uniqueness from [12] (hereTmax is the maximal existence time for the flow and may equal ∞). Notethat if Cg =∞, then the integral curvature pinching condition (1.5) implies|Rm|g ≡ 0 onM and subsequently the conclusion is trivially satisfied by thestationary solution g(t) ≡ g. So without loss of generality, we may assumeCg < ∞ in which case we will prove the theorem assuming that the curva-ture pinching inequality (1.5) is strict, albeit with the slightly larger constantδ1(n) replacing δ(n) as defined in Remark 1.2.2. Here we will assume thatδ1(n) still satisfies the condition in (1.6), that is δ1(n) < (40nL(n))−1For ease of reference, we summarize our assumptions as follows:(Mn, g(t)), n ≥ 3, t ∈ [0, Tmax), g(0) = g, is a completeRicci flow, and for any T < Tmax, supM×[0,T ]|Rm|(x, t) <∞.(∫ |Rm|n2 dVg) 2n < δ1(n) 1Cg where Sob(M, g) ≤ [Cg, 0].(3.6)We are now ready to continue with the proof.Our first step is to determine the evolution of Lp norm of the curvature.We choose to break up into two different cases based on the value of p becausethe proof is simpler when p/2 ≥ 1 and may be of independent interest.Lemma 3.2.1. Let (Mn, g(t)), t ∈ [0, T ] be a complete bounded curvatureRicci flow and fix some smooth function ϕ : M → R with compact support.Then the following hold.50(a) If α ≥ 1, thenddt∫ϕ2|Rm|2α dVt ≤ −2∫|∇(ϕ|Rm|α)|2 dVt+ (16α+ n)∫ϕ2|Rm|2α+1 dVt + 2∫|∇ϕ|2|Rm|2α dVt.(b) If 12 < α < 1, then for any β > 0, there holdsddt∫ϕ2(|Rm|2 + β)α dVt ≤ −C1(α)∫|∇(ϕ(|Rm|2 + β)α2 )|2 dVt+ (16α+ n)∫ϕ2(|Rm|2 + β)α+ 12 dVt+ C2(α)∫|∇ϕ|2(|Rm|2 + β)α dVt.Moreover, C1(α) → 0 as α → 12 and if 23 < α < 1, then one can takeC1(α) =12 .Proof. First recall the well-known evolution of |Rm|2 under the Ricci flow:∂t|Rm|2 ≤ ∆|Rm|2 − 2|∇Rm|2 + 16|Rm|3. (3.7)For part (a), we combine this with ∂t dVt = −Rt dVt, the Kato inequality|∇|T || ≤ |∇T |, inequality (1.19) |R| ≤ n|Rm| (see (1.19)), and α ≥ 1 whichgivesddt∫ϕ2|Rm|2α dVt=∫ϕ2∂∂t(|Rm|2α dVt)=∫ϕ2(α|Rm|2(α−1)∂t|Rm|2 −R|Rm|2α)dVt≤∫ϕ2α|Rm|2(α−1) (∆|Rm|2 − 2|∇Rm|2 + 16|Rm|3)− ϕ2R|Rm|2α dVt≤∫ϕ2[α|Rm|2(α−1)∆|Rm|2 − 2α|Rm|2α−2|∇|Rm||2+ (16α+ n) |Rm|2α+1]dVt=∫ϕ2α|Rm|2(α−1)∆|Rm|2 dVt − 2α∫ϕ2|∇|Rm|α|2 dVt+ (16α+ n)∫ϕ2|Rm|2α+1 dVt.51Now we integrate by parts the Laplacian term using that the support of ϕis compact (so that no boundary integrals are introduced) to obtain∫ϕ2α|Rm|2(α−1)∆|Rm|2 dVt= −∫α∇(ϕ2|Rm|2(α−1))∇|Rm|2 dVt= −∫α∇ϕ2|Rm|2(α−1)∇|Rm|2 dVt −∫αϕ2∇|Rm|2(α−1)∇|Rm|2 dVt= −2∫∇ϕ2|Rm|α∇|Rm|α dVt −∫αϕ2∇|Rm|2(α−1)∇|Rm|2 dVt= −4∫ϕ∇ϕ|Rm|α∇|Rm|α dVt − α∫ϕ2∇|Rm|2(α−1)∇|Rm|2 dVt= I− α II.The first term above is simplified by expanding out the derivativeI =− 2∫|∇(ϕ|Rm|α)|2 dVt+ 2∫|∇ϕ|2|Rm|2α dVt+ 2∫ϕ2|∇|Rm|α|2 dVt.(3.8)For the second term, we have to be careful because ∇|Rm|2(α−1) is singularat points where |Rm| = 0 when α < 3/2 (though this singularity is counteredby a contribution of |Rm| from the ∇|Rm|2 term). Since α ≥ 1, for any ϵ > 0we have∇(|Rm|+ ϵ)2(α−1)∇(|Rm|+ ϵ)2= 4(α− 1)(|Rm|+ ϵ)2α−3(|Rm|+ ϵ)|∇(|Rm|+ ϵ)|2= 4(α− 1)(|Rm|+ ϵ)2α−2|∇(|Rm|+ ϵ)|2=4(α− 1)α2|∇(|Rm|+ ϵ)α|2.Of course the right-hand side converges to 4(α−1)α2|∇|Rm|α|2 pointwise asϵ → 0, and since ϕ is compactly supported we can apply the Dominated52Convergence Theorem (twice) to see thatII = limϵ→0∫ϕ2∇(|Rm|+ ϵ)2(α−1)∇(|Rm|+ ϵ)2 dVt= limϵ→04(α− 1)α2∫ϕ2|∇(|Rm|+ ϵ)α|2 dVt=4(α− 1)α2∫ϕ2|∇|Rm|α|2 dVt.(3.9)So using (3.8) and (3.9), we have∫ϕ2α|Rm|2(α−1)∆|Rm|2 dVt= −4∫ϕ∇ϕ|Rm|α∇|Rm|α dVt − 4(α− 1)α∫ϕ2|∇|Rm|α|2 dVt.Finally, we gather all the ϕ2|∇|Rm|α|2 terms to see that[− 2α− 4(α− 1)α+ 2] ∫ϕ2|∇|Rm|α|2 dVt= −2(α− 1)α∫ϕ2|∇|Rm|α|2 dVt ≤ 0,(3.10)where we have again used that α ≥ 1. This proves part (a).(b) The proof is similar to [4, Lemma 2.1], though in that proof it is assumedthat α ≥ n4 and that ϕ is a function of space and time. Since our conditionon α is different and our ϕ is a function of space only, we provide the detailsfor the readers’ convenience and to keep track of the constants. We beginjust as before:ddt∫ϕ2(|Rm|2 + β)α dVt=∫ϕ2(α(|Rm|2 + β)α−1∂t|Rm|2 −R(|Rm|2 + β)α)dVt≤∫ϕ2α(|Rm|2 + β)α−1 (∆|Rm|2 − 2|∇|Rm||2)+ (n+ 16α)ϕ2(|Rm|2 + β)α+ 12 dVt.53Now we integrate by parts the Laplacian term which gives∫ϕ2α(|Rm|2 + β)α−1∆|Rm|2 dVt= −∫α∇ (ϕ2(|Rm|2 + β)α−1)∇|Rm|2 dVt= −4α∫ϕ⟨∇ϕ,∇|Rm|⟩(|Rm|2 + β)α−1|Rm| dVt− 4α(α− 1)∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt.Note that the coefficient in the latter term is actually positive because α < 1,so we have− 4α(α− 1)∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt≤ −4α(α− 1)∫ϕ2(|Rm|2 + β)α−1|∇|Rm||2 dVt.For the former term, we apply Cauchy-Schwarz, Ho¨lder’s, and Young’s in-equalities− 4α∫ϕ⟨∇ϕ,∇|Rm|⟩(|Rm|2 + β)α−1|Rm|dVt≤ 4α∫ϕ|∇ϕ| |∇|Rm|| (|Rm|2 + β)α−1|Rm|dVt≤ 4α∫ϕ|∇ϕ| |∇|Rm|| (|Rm|2 + β)α− 12 dVt≤ 4α(∫ϕ2(|Rm|2 + β)α−1|∇|Rm||2 dVt) 12(∫|∇ϕ|2(|Rm|2 + β)α dVt) 12≤ 2αϵ∫ϕ2(|Rm|2 + β)α−1|∇|Rm||2 dVt + 2αϵ∫|∇ϕ|2(|Rm|2 + β)α dVt= I+ II.Now we can gather up the terms in the form of I and restrict ϵ > 0 smallenough so that the coefficient is negative. Then we can use Cauchy-Schwarz54again (but in reverse) to see that[−2α+ 4α(1− α) + 2αϵ]∫ϕ2(|Rm|2 + β)α−1|∇|Rm||2 dVt≤ [−2α+ 4α(1− α) + 2αϵ]∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt= [−2α+ 4α(1− α) + 2αϵ] 1α2∫ϕ2|∇(|Rm|2 + β)α2 |2 dVt≤ [−2α+ 4α(1− α) + 2αϵ] 1α2(12∫|∇ϕ(|Rm|2 + β)α2 |2 dVt−∫|∇ϕ|2(|Rm|2 + β)α dVt).Now whenever α > 12 , we can restrict ϵ small enough so that−C1(α) := [−2α+ 4α(1− α) + 2αϵ] 12α2< 0,and if α > 23 , ϵ can be taken small enough so that C1(α) =12 . This provesthe result.We now prove that our hypotheses of (3.6) are enough to ensure thatthe scale-invariant integral curvature quantity∫ |Rm|n2 dVt is nonincreasingin time. The following Lemma 3.2.2 is similar to [6, Lemma 2.2]. In fact thenecessary tools needed in both proofs are the same:(a) A weighted Sobolev inequality with scalar curvature term along theflow; and(b) smallness of the scale-invariant integral curvature∫ |Rm|n2 dVt uni-formly along the flow relative to the constant in the Sobolev inequal-ity.In [6] when n ≥ 4, both (a) and (b) are guaranteed by a rescaling argu-ment from the short-time existence result in [5]. For us however, we obtain(a) from Theorem 2.4.1 while (b) is guaranteed for some short time intervalby the upper-continuity of t 7→ ∫ |Rm|n2 dVt from Proposition 3.1.2, whichwill be sufficient for our proof below. Since our hypotheses and some lines ofargument differ from those in [6], we will include the proof for completeness.We will also refer to these steps in establishing the case for n = 3, and aswell as Lemma 3.2.4 which extends this monotonicity result to other powersof |Rm|.55Lemma 3.2.2. Let (Mn, g(t)) satisfy the conditions in (3.6). Then t 7→∫ |Rm|n2 dVt is nonincreasing on [0, Tmax).Proof. We first prove the result for dimensions n ≥ 4 and comment on thenecessary modification for dimension n = 3 at the end. Fix some T < Tmax.It clearly suffices to show that t 7→ ∫ |Rm|n2 dVt is nonincreasing on [0, T ].Let ϕ : M → R be some smooth compactly supported function to bechosen later. Lemma 3.2.1 part (a) (applied with α = n4 ) givesddt∫ϕ2|Rm|n2 dVt ≤ −2∫|∇(ϕ|Rm|n4 )|2 dVt+(4n+ n)∫ϕ2|Rm|n2+1 dVt + 2∫|∇ϕ|2|Rm|n2 dVt.Note that δ1(n) <2n so that the conclusion of Theorem 2.4.1 holds withA = L(n)Cg, thus giving− 2∫|∇(ϕ|Rm|n4 )|2 dVt≤ − 2A(∫(ϕ|Rm|n4 ) 2nn−2 dVt)n−2n+ 2∫R+ϕ2|Rm|n2 dVt≤ − 2A(∫(ϕ|Rm|n4 ) 2nn−2 dVt)n−2n+ 2n∫ϕ2|Rm|n2+1 dVt.Now we gather the terms with |Rm|n2+1 in the integrand and apply Ho¨lder’sinequality to obtain7n∫ϕ2|Rm|n2+1 dVt ≤ 7n(∫|Rm|n2 dVt) 2n(∫ϕ2nn−2 |Rm|n4 2nn−2 dVt)n−2n.To summarize the above, for any compactly supported smooth ϕ we haveddt∫ϕ2|Rm|n2 dVt≤(∫ϕ2nn−2 |Rm|n4 2nn−2 dVt)n−2n[− 2A+ 7n(∫|Rm|n2 dVt) 2n]+ 2∫|∇ϕ|2|Rm|n2 dVt.(3.11)56The choice of ϕ we will be employing is as follows. Let ψ : R→ [0, 1] bea smooth decreasing cutoff function which satisfies ψ ≡ 1 on (−∞, 1] andψ ≡ 0 on [2,∞). Fix some x0 ∈M and define, for r > 0,ϕr(x) = ψ(d˜0(x)r).Here d˜0 is smooth with uniformly bounded gradientsupx∈M|∇d˜0|g ≤ c,and uniformly bounded from below by the distance function from x0 ∈ Mrelative to g0 in the sense that c−1dg0(x0, x) ≤ d˜0(x) for all x ∈M for somec > 0 depending on g0 (in particular on a lower bound of Rcg0 , see [38,Theorem 4.2]). From this, assumption (3.6), and Proposition 3.1.2, we candefine the following constants which may depend on T < Tmax :K := supM×[0,T ]|Rm| <∞β := supt∈[0,T ]∫|Rm|n2 dVt <∞C := supr≥1supM×[0,T ]r2|∇g(t)ϕr|2g(t) <∞.Now we define the functionF (r, t) :=∫ϕ2r |Rm|n2 dVtwhere r ≥ 1 and t ∈ [0, T ] and defineT ′ := sup{t ∈ [0, T ] : F (∞, t) ≤ (δ1/Cg)n2 } > 0where the non-emptyness of the set on the right-hand side follows fromassumption (3.6) while the positivity of T ′ follows from Proposition 3.1.2(in particular, the upper-continuity of t 7→ ∫ |Rm|n2 dVt).Now using the fact that [− 2A + 7n δ1Cg(0) ] < 0, which follows from A =L(n)Cg(0), and our choice of δ1 in assumption (3.6) as well as the definitionof C, β above, the evolution (3.11) then givesddtF (r, t) ≤ 2 Cβr257for all r ≥ 1 and all t ∈ [0, T ′]. In particular, for all 0 < s1 < s2 < T ′ thisgives ∫|Rm|n2 dVs2 = limr→∞F (r, s2)≤ limr→∞(F (r, s1) + 2C(s2 − s1)βr2)= F (∞, s1) =∫|Rm|n2 dVs1so that∫ |Rm|n2 dVt is non-increasing for t ∈ [0, T ′]. Here we have used thatd˜0(x) ≥ c−1dg(x, x0) and thus {ϕr = 1} exhausts M as r → ∞, as well asseveral applications of Dominated Convergence. But since∫ |Rm|n2 dV0 <(δ1/Cg(0))n2 , this just imples that T ′ = T so that∫ |Rm|n2 dVt is nonincreas-ing for t ∈ [0, T ]. This concludes the proof of the lemma for n ≥ 4.For the dimension n = 3 case, we proceed just as we did above withpart (b) of Lemma 3.2.1 instead of part (a). The analogue of the evolutioninequality (3.11) that we obtain in this case isddt∫ϕ2(|Rm|2 + β)α dVt≤ C2(3/4)∫|∇ϕ|2(|Rm|2 + β)α dVt +(∫ϕ6(|Rm|2 + β) 94 dVt) 13×[− 12A+(15 +32)(∫suppϕ(|Rm|2 + β) 34 dVt) 23].Now for fixed r ≫ 0 and β > 0, we can integrate the above inequality onthe interval [t0, t1], then send β → 0 (which is validated by the presence ofthe cutoff function ϕ) to obtain∫ϕ2|Rm| 32 dVt1 −∫ϕ2|Rm| 32 dVt0≤ C2(3/4)∫ t1t0∫|∇ϕ|2|Rm| 32 dVt dt+∫ t1t0(∫ϕ6|Rm| 92 dVt) 13×[− 12A+(15 +32)(∫suppϕ|Rm| 32 dVt) 23]dt.(3.12)58From here, one can show t 7→ ∫ |Rm| 32 dVt is nonincreasing just as in thehigher dimensional case above. In the process, we use the fact thatδ1(3) <133L(3)as outlined in (1.6), so that the bracketed term in (3.12) is nonpositive.The following corollary is not essential to our proof (any time it is usedwe could use Theorem 2.4.1 and Lemma 3.2.2 instead) but we include itsstatement for ease of reference. The proof is identical to the proof of [13,Corollary 4.3].Corollary 3.2.3. Let (Mn, g(t)) satisfy the conditions in (3.6). Then theuniform Sobolev inequalitySob(M, g(t)) ≤ [A0, 0]holds for all t ∈ [0, Tmax). Moreover we can take A0 = 2L(n)Cg0.Lemma 3.2.4. Let (Mn, g(t)) satisfy the conditions in (3.6). Then t 7→∫ |Rm|q dVt is nonincreasing on [0, Tmax) for any q satisfying max{2, n2 } ≤q ≤ n2 nn−2 .Proof. We proceed in a similar fashion as Lemma 3.2.2. Fix T < Tmax andnote that Lemma 3.2.2 combined with our bounded curvature assumptionimpliessupt∈[0,T ]∥Rm∥Lqg(t)<∞.We apply Lemma 3.2.1 part (a) with α = q/2 (note that α ≥ 1) and Ho¨lder’sinequality to seeddt∫ϕ2|Rm|2α dVt≤ −2∫|∇(ϕ|Rm|α)|2 dVt + 2∫|∇ϕ|2|Rm|2α dVt+ (16α+ n)∫ϕ2|Rm|2α+1 dVt≤ −2∫|∇(ϕ|Rm|α)|2 dVt + 2∫|∇ϕ|2|Rm|2α dVt+ (16α+ n)(∫|Rm|n2 dVt) 2n(∫ϕ2nn−2 |Rm|α 2nn−2 dVt)n−2n.59Now Corollary 3.2.3 gives−2∫|∇(ϕ|Rm|α)|2 dVt ≤ − 2A0(∫ϕ2nn−2 |Rm|α 2nn−2 dVt)n−2n.We will also retain from Lemma 3.2.2 the notation of ϕr and the constantC for whichsup(x,t)∈M×[0,T ]|∇g(t)ϕr|2g(t) ≤Cr2.Of course, the constant C here will depend on the bound of the curvatureonM× [0, T ], but not on r. Then plugging in A0 = 2LCg0 , applying Lemma3.2.2, and using our definition of δ1(n) yieldsddt∫ϕ2r |Rm|q dVt≤(∫ϕ2nn−2r |Rm|qnn−2 dVt)n−2n[− 22LCg0+ (8q + n)(∫|Rm|n2 dVt) 2n]+ 2∫|∇ϕr|2|Rm|2α dVt≤(∫|Rm|q nn−2 dVt)n−2n[− 1LCg0+(4n2n− 2 + n)δ1(n)Cg0]+ 2Cr2∫|Rm|2α dVt≤ 2Cr2supt∈[0,T ]∫|Rm|q dVt,where we have used that4n2n− 2 + n ≤ 13nfor n ≥ 3 as well as our definition of δ1(n). Thus for fixed 0 ≤ t0 < t1 ≤ T ,we may integrate the above from t0 to t1 to see∫ϕ2r |Rm|q dVt1 ≤∫ϕ2r |Rm|q dVt0 + 2(t1 − t0)Cr2supt∈[0,T ]∫|Rm|q dVt≤∫|Rm|q dVt0 + 2(t1 − t0)Cr2supt∈[0,T ]∫|Rm|q dVt.Taking r →∞ completes the proof.60Proposition 3.2.5. Let (Mn, g(t)) satisfy the conditions in (3.6). Thenthe curvature estimatesupx∈M|Rm|2(x, t) ≤ max(c1(g0)t−2(n−2)n , c2(g0)t4(n−2)n2 )holds for all t ∈ [0, Tmax). Here c1(g0) and c2(g0) are positive constantsdepending only on the initial metric g0. In particular, they depend only onCg0 and the initial curvature quantity∫ |Rm|n2 nn−2 dV0.Proof. Like the previous lemmas, we will fix some T < Tmax and prove theestimate on [0, T ]. For convenience of notation, let f(x, t) = |Rm|2(x, t) andwrite ϕ to be some ϕr just as in the notation used in Lemma 3.2.2 (we’llspecify r later). By Lemma 3.2.1, for any p ≥ n4 nn−2 (note that this conditionforces p ≥ 1), there holdsddt∫ϕ2fp dVt +∫|∇(ϕf p2 )|2 dVt≤ −∫|∇(ϕf p2 )|2 dVt + (16p+ n)∫ϕ2|Rm|fp dVt + 2∫|∇ϕ|2fp dVt.=: I+ (16p+ n)II+ III.(3.13)We will estimate the above terms I, II, III separately.Corollary 3.2.3 immediately yieldsI ≤ − 1A0(∫ϕ2nn−2 fpnn−2 dVt)n−2n(3.14)where A = 2L(n)Cg0 . To estimate II, we recall the generalized Ho¨lder’sinequality: ∥∥∥∥∥∥k∏j=1fj∥∥∥∥∥∥L1≤k∏j=1∥fj∥Lrj wherek∑j=11ri= 1.We will apply this withf1 = |Rm|f2 = η−(1− n2q)n2qϕ2(1− n2q)fp(1− n2q)f3 = η(1− n2q)n2qϕnq fp n2qr1 = qr2 =11− n2qr3 =2qnnn−261where q = n2nn−2 and η > 0 will be chosen later. This givesII ≤(∫|Rm|q dVt) 1q(η− n2q∫ϕ2fp dVt)1− n2q×(η(1− n2q)nn−2∫ϕ2nn−2 fpnn−2 dVt)n−2nn2q.Now we apply Lemma 3.2.4, which states that t 7→ ∥Rm∥Lqg(t)is nonincreas-ing along the flow. So writing β = ∥Rm∥Lqg(0), we have that ∥Rm∥Lqg(t)≤ βfor all t ≥ 0. Now to the remaining terms we apply Young’s inequalityab ≤ ar1r1+br2r2, where1r1+1r2= 1,with r1 =11− n2q, r2 =2qn to see thatII ≤ β(η− n2q∫ϕ2fp dVt)1− n2q(η(1− n2q)nn−2∫ϕ2nn−2 fp22nn−2 dVt)n−2nn2q≤ β(1− n2q)η− n2q∫ϕ2fp dVt + βn2qη1− n2q(∫ϕ2nn−2 fpnn−2 dVt)n−2n.(3.15)Finally, to estimate III, we can use Lemma 3.2.2 combined with the factthat (M, g(t)), t ∈ [0, T ] is a bounded curvature flow to obtainsupt∈[0,T ]∫fp dVt <∞. (3.16)Then following just as in the proof of the previous lemmas, for any ϵ > 0,there exists rϵ ≫ 0 such that whenever r ≥ rϵ and t ∈ [0, T ] there holdsIII < ϵ. (3.17)Putting (3.14), (3.15) and (3.17) back into (3.13) we haveddt∫ϕ2fp dVt +∫|∇(ϕf p2 )|2 dVt≤ β(16p+ n)(1− n2q)η− n2q∫ϕ2fp dVt+[βn(16p+ n)2qη1− n2q − 1A0](∫ϕ2nn−2 fpnn−2 dVt)n−2n+ ϵ.62Now we choose η > 0 so that the bracketed quantity above is zero.Precisely, we takeη1− n2q =2qβn(16p+ n)A0,so that plugging this value of η back into the above, and simplifying givesddt∫ϕ2fp dVt +∫|∇(ϕf p2 )|2 dVt≤ β(16p+ n)(1− n2q)(2qβn(16p+ n)A0)− 2q2q−nn2q∫ϕ2fp dVt + ϵ= β(16p+ n)2n(A0(n− 2)β(16p+ n)n)n−22∫ϕ2fp dVt + ϵ≤ (20p)n2 β n2An−220∫ϕ2fp dVt + ϵ =: Kp∫ϕ2fp dVt + ϵ,(3.18)where in the second line we used our value of q = n2nn−2 , and in the last linewe used the estimates 16p + n ≤ 20p and n ≥ 3. Now for 0 < τ < τ ′ ≤ T ,define ψ : [0, T ]→ [0, 1] byψ(t) =0 0 ≤ t ≤ τt−ττ ′−τ τ ≤ t ≤ τ ′1 τ ′ ≤ t ≤ T.Multiply (3.18) by ψ and find thatddt(ψ∫ϕ2fp dVt)+ ψ∫|∇(ϕf p2 )|2 dVt ≤ (Kpψ + ψ′)∫ϕ2fp dVt + ϵ,so that after integrating from τ to t˜ ∈ [τ ′, T ], we obtain∫ϕ2fp dVt˜ +∫ t˜τ ′∫|∇(ϕf p2 )|2 dVt dt≤ ψ(t˜)∫ϕ2fp dVt˜ − ψ(τ)∫ϕ2fp dVτ +∫ t˜τψ∫|∇(ϕf p2 )|2 dVt dt=∫ t˜τ[ddt(ψ∫ϕ2fp dVt)+ ψ∫|∇(ϕf p2 )|2 dVt]dt≤∫ t˜τ(Kpψ + ψ′)∫ϕ2fp dVt dt+ (t˜− τ)ϵ≤(Kp +1τ ′ − τ)∫ Tτ∫fp dVt dt+ Tϵ.63In particular, because both terms on the left-hand side are nonnegative, theabove yields two inequalities which will be used later. The first is∫ Tτ ′∫|∇(ϕf p2 )|2 dVt dt≤(Kp +1τ ′ − τ)∫ Tτ∫fp dVt dt+ Tϵ for any ϵ > 0, r ≥ rϵ.(3.19)Likewise, considering the other term on the left-hand side gives∫ϕ2fp dVt˜ ≤(Kp +1τ ′ − τ)∫ Tτ∫fp dVt dt+ Tϵfor any t˜ ∈ [τ ′, T ]. Since the integrand ϕ2fp is nonnegative however, wecan take r → ∞, followed by taking ϵ → 0 which gives the second usefulinequality: ∫fp dVt˜ ≤(Kp +1τ ′ − τ)∫ Tτ∫fp dVt dt. (3.20)For p ≥ 1 and τ ∈ [0, T ], we defineH(p, τ) =∫ Tτ∫fp dVt dt,and let ν = 1 + 2n . We need to prove the following claim:Claim 3.2.6. For 0 ≤ τ < τ ′ ≤ T , there holdsH(νp, τ ′) ≤ A0(Kp +1τ ′ − τ)νH(p, τ)ν .Proof of Claim: We begin by assuming there is a ϕ2 term inside the inte-grand in the left-hand side. Then by applying Ho¨lder’s inequality followedby Corollary 3.2.3, we obtain∫ Tτ ′∫ϕ2fp(1+2n) dVt dt≤∫ Tτ ′(∫suppϕfp dVt) 2n(∫(ϕfp2 )2nn−2 dVt)n−2ndt≤ A0(supτ ′≤t≤T∫suppϕfp dVt) 2n ∫ Tτ ′∫|∇(ϕf p2 )|2 dVt dt.64Applying (3.19), and noting that the supremum above is achieved at somet˜ ∈ [τ ′, T ] (of course, the time t˜ possibly depends on ϕ), we can continue ourestimate above to∫ Tτ ′∫ϕ2fp(1+2n) dVt dt≤ A0(∫fp dVt˜) 2n[(Kp +1τ ′ − τ)∫ Tτ∫fp dVt dt+ Tϵ]≤ A0(∫fp dVt˜) 2n(Kp +1τ ′ − τ)H(p, τ) +A0Tϵ(∫fp dVt˜) 2n≤ A0(Kp +1τ ′ − τ)νH(p, τ)ν +A0Tϵ(supt∈[0,T ]∫fp dVt) 2n.Now the right-hand side is independent of r ≥ rϵ, so we can take r → ∞,apply the Monotone Convergence Theorem, then take ϵ→ 0. Note that thequantity supt∈[0,T ]∫fp dVt is finite by (3.16) above. This proves the claim.At this point, the remainder of the proof is verbatim to [13, Proposition4.5]. We will however provide an outline to compute the constants involvedand for our future reference. Let p0 =q2 =n4nn−2 , recall ν = 1+2n and defineη = νn2 , pk = νkp0, τk = (1− η−k)T, Φk = H(pk, τk)1pk .So applying Claim 3.2.6, we getΦk+1 ≤ A1νpk0(βn2An−220 (20p0)n2 +1(1− η−1)T) 1pkηkpkΦkand iterating the above giveslimk→∞Φk ≤ A1ν∞∑k=01pk0(βn2An−220 (20p0)n2 +1(1− η−1)T) ∞∑k=01pkη∞∑k=0kpkΦ0.(3.21)65We can also calculate∞∑k=01pk=2(n2 − 4)n2,1ν∞∑k=01pk=2(n− 2)n,∞∑k=0kpk=n2 − 4n.Finally,ηn2−4n = νn2n2−4n =(1 +2n)n2−42≤ en−1because the function(1 + 2x)x2−42 e1−x is increasing on [0,∞) and boundedby 1. Plugging these identities and p0 =n4nn−2 ≤ n back into (3.21) giveslimk→∞Φk ≤ en−1A2(n−2)n0(βn2An−220 (20n)n2 +1(1− η−1)T) 2(n2−4)n2Φ0. (3.22)Just as in [13], there holdsΦ0 ≤ T4(n−2)n2 β2 and limk→∞Φk ≥ ∥f∥L∞g(T ).Using this in (3.22) givessupx∈M|Rm|2g(T ) = ∥f∥L∞g(T )≤ en−1A2(n−2)n0 β2T4(n−2)n2(βn2An−220 (20n)n2 +1(1− η−1)T) 2(n2−4)n2≤ 2 2(n2−4)n2 en−1A2(n−2)n0 β2T4(n−2)n2 ×max(βn2An−220 (20n)n2) 2(n2−4)n2,(1(1− η−1)T) 2(n2−4)n2 .Because T > 0 was arbitrary, this completes the proof.66Corollary 3.2.7. Let (Mn, g(t)) satisfy the conditions in (3.6). Then theflow exists for all times, i.e., Tmax =∞, andsupM×[0,∞)|Rm|(x, t) <∞.Proof. The proof is identical to [13, Corollary 4.6].Lemma 3.2.8. Let (Mn, g(t)) satisfy the conditions in (3.6). Then for anyγ > 0, there exists Tγ > 0 such that for all t ≥ Tγ, there holds(∫|Rm|n2 nn−2dVt)n−2n≤ γt−1.Proof. Start by fixing T > 0. Recall that in Lemma 3.2.2, we proved thatfor any ϵ > 0, there exists rϵ ≫ 0 such that whenever r ≥ rϵ, there holds∫ϕ2r |Rm|n2 dVT −∫ϕ2r |Rm|n2 dV0≤ − 110A∫ T0(∫ϕ2nn−2r |Rm|n42nn−2 dVt)n−2ndVt + Tϵ.Rearranging this in turn gives∫ T0(∫ϕ2nn−2r |Rm|n2nn−2dVt)n−2ndt≤ 10A[∫ϕ2r |Rm|n2 dV0 −∫ϕ2r |Rm|n2 dVT]+ 10TAϵ≤ 10A∫ϕ2r |Rm|n2 dV0 + 10TAϵ ≤ 10A∫|Rm|n2 dV0 + 10TAϵ.(3.23)Now take r →∞, then ϵ→ 0, then T →∞ to see that∫ ∞0(∫M|Rm|n2 nn−2dVt)n−2ndt ≤ 10A∫|Rm|n2 dV0 <∞.This combined with the fact that t 7→ ∫ |Rm|n2 nn−2 dt is nonincreasing (Lemma3.2.4) giveslimt→∞ t(∫|Rm|n2 nn−2 dVt)n−2n= 0.67Proposition 3.2.9. Let (Mn, g(t)) satisfy the conditions in (3.6). Thenthe curvature estimate (b) holds.Proof. The proof is the same as [13, Proposition 4.8].At this point, we diverge from the layout from [13] to prove the othercurvature estimate and the convergence statement.Proposition 3.2.10. Let (Mn, g(t)) satisfy the conditions in (3.6). Thenthe curvature estimate (a) holds.Proof. By slightly modifying the proof of Proposition 3.2.5 (see also [13,Proposition 4.8]) with the only changes beingτk =T2+ (1− η−k)T2, and β = supt∈[T/2,T ]∥|Rm|∥Lqg(t),we obtain the estimatesupx∈M|Rm|g(t) ≤ 2n2−4n2 e12(n−1)An−2n0 βt2(n−2)n2 ×max(βn2An−220 (20n)n2)n2−4n2,(2(1− η−1)t)n2−4n2 .(3.24)Note that Lemma 3.2.4 implies that t 7→ ∫ |Rm|q dVt is nonincreasing, so wemay take β = ∥|Rm|∥Lqg(T/2). Now by slightly modifying the proof of Lemma3.2.8 (specifically taking r →∞ then ϵ→ 0 in inequality (3.23)), we havet2βn2 ≤∫ t/20(∫|Rm|n2 nn−2 dVs)n−2nds≤ 10A∫|Rm|n2 dV0≤ 10L(n)Cg(δ1(n)Cg)n2.Then since A0 = 2L(n)Cg, we can see thatAn−2n0 β ≤ (2LCg)n−2n (10LCg)2n22n δ1(n)Cgt2n≤ 14nt2n.68Plugging this back into our curvature bound (3.24) we havesupx∈M|Rm|g(t) ≤ 2n2−4n2 e12(n−1) 14nt2nt2(n−2)n2 ×max((20n)n2(4n)n2 t)n2−4n2,(2(1− η−1)t)n2−4n2< (5e)n2 t−2n−1+ 4n2+ 2n− 4n2 =(5e)n2t=C0(n)t,where in the latter inequality we used the fact that2(1− η−1) ≤ 4 ≤ 5n2 .Now we prove the metric convergence and topological rigidity statement,under an additional Lp-curvature condition.Proposition 3.2.11. Let (Mn, g(t)) satisfy the conditions in (3.6) and fur-thermore assume that |Rm|g ∈ Lp for some p < n2 . Then g(t) convergeslocally smoothly to a complete flat limit metric g(∞) on Mn with Euclideanvolume growth, and thus Mn is diffeomorphic to Rn.Proof. In this proof, we will use C(·, ·, · · · ) to denote a constant which de-pends on its arguments, and may differ line-by-line. First note that byinterpolation, we may assume that{23 <p2 <n4 n = 3, 4n−24 <p2 <n4 n ≥ 5.(3.25)For such a fixed value of p (for each n), write q = nn−2p. Now we note thatin establishing (3.18) in the proof of Proposition 3.2.5, the only assumptionson p˜, q˜ there were p˜ ≥ 1 and max{2, n/2} ≤ q˜ ≤ n2 nn−2 so that Lemma 4.4 inturn guaranteed that t 7→ ∥|Rm|∥Lq˜g(t)is nonincreasing. Thus for any p˜ ≥ 1and our choice of q above, we haveddt∫ϕ2f p˜ dVt +14∫|∇(ϕf p˜2 )|2 dVt≤ 4β(16p˜+ n)(1− n2q)(βn(8p˜+ n)A02q) n2q−n∫ϕ2f p˜ dVt + ϵ= (p˜β)2q2q−nC(n, q, Cg)∫ϕ2f p˜ dVt + ϵ.69Now we continue on in identical fashion to the proof of Proposition 3.2.5(in going from (3.18) to (3.21)) with the only modifications being that wetakep˜0 =q2, η = ν2q2q−n , τk =T2+ (1− η−k)T2, β = supt∈[T/2,T ]∥|Rm|∥Lqg(t).In light of our choice of p, q and Lemma 3.2.4, we in fact can take β =∥|Rm|∥Lqg(T/2). Also, by our choice of p and q, we have that p˜0 > 1. Theresult after iterating issupx∈M|Rm|(x, T )2 ≤ C(n, p, Cg)(β2q2q−n + T−1) ∞∑k=01p˜k Φ0.Also, in this case we have∞∑k=01p˜k=1p˜0∞∑k=01(1 + 2n)k=n+ 2q.Now recalling the definition of Φ0 and the fact that∫M |Rm|q dVt is nonin-creasing in t, we haveΦ0 = H(q/2, τ0)2q =(∫ TT/2∫M|Rm|q dVt dt) 2q= C(q)T2q ∥|Rm|∥2Lqg(T/2).We need to control the value of ∥|Rm|∥Lqg(T/2). For this, we proceed justas in Lemma 3.2.2 and 3.2.4 with α = p2 (here we use the assumption thatp2 >23 so that we can apply part (b) of Lemma 3.2.1) to obtain−∫|Rm|p dV0 ≤∫ t0(∫M|Rm|q dVs)n−2nds[− 14A+ (16α+ n)δ1(n)Cg]≤∫ t0(∫M|Rm|q dVs)n−2nds[− 14A+5n40nA]≤ − 110A∫ t0(∫M|Rm|q dVs)n−2nds ≤ − t10A∥|Rm|∥qn−2nLqg(t).Of course, this argument relies on the fact that t 7→ ∫ |Rm|p dVt is uniformlybounded on finite time intervals which is true by Proposition 3.1.2 because70our assumption (3.25) in particular implies p > 1. So rearranging the abovegives∥|Rm|∥qn−2nLqg(t)≤ C(n, p, Cg, ∥|Rm|∥Lpg0 )1twhich impliesΦ0 ≤ C(n, p, Cg, ∥|Rm|∥Lpg0 )T(− nn−2+1) 2q= C(n, p, Cg, ∥|Rm|∥Lpg0 )T− 4q(n−2) ,andβ2q2q−n ≤ C(n, p, Cg, ∥|Rm|∥Lpg0 )T− 2n(n−2)(2q−n) .Note that2n(n− 2)(2q − n) ≥ 1 ⇐⇒ q ≤n22(n− 2)and the latter is true by our choice of q. Therefore for T large, we havesupx∈M|Rm|2(x, T ) ≤ C(n, p, Cg, ∥|Rm|∥Lpg0 )T− n2q(n−2) (3.26)But q < n2nn−2 and so we have that supx∈M |Rm|g(t) ≲ t−(1+σ) for someσ > 0. Then by Shi’s estimates [40] (see also [32, Theorem 4.7]), we havethat for all k ≥ 0, there holdssupx∈M|∇kRm|g(t) ≤ Ckt−1−σ−k/2 (3.27)for all t ∈ (0,∞) and some Ck(n, p, Cg, ∥|Rm|∥Lpg0 ) > 0.We now show that the integrability in t of the right-hand side of (3.27)implies that g(t) converges in C∞loc(M) as t → ∞ to a complete flat metricg(∞) on M that is equivalent to g(0). This is by a standard Ricci flowargument which we now sketch. In the proceeding paragraphs, the constantsC,Ci, ci can be taken to depend only on n, p, Cg, ∥|Rm|∥Lpg0 .Let V denote an arbitrary unit vector field on M with respect to themetric g(1). Then for any t ≥ 1 we havelog ∥V ∥2g(t) = log ∥V ∥2g(t) − log ∥V ∥2g(1)=∫ t1ddtlog ∥V ∥2g(t) dt= 2∫ t1Rcg(t)(V, V )∥V ∥2g(t)dt71while the integrand on the right-hand side is bounded in absolute value byn supM |Rm|g(t) and thus C/t1+σ for some σ > 0 by (3.27) with k = 0.It follows that log ∥V ∥2g(t) converges - uniformly in space - as t → ∞ toa bounded function on M . From this we conclude that g(t) converges inL∞(M) as t → ∞ to a metric g(∞) which is uniformly equivalent to g(0),and thus also complete.We show that in fact g(t)→ g(∞) in C∞loc(M) and thus g(∞) is a smoothflat metric onM equivalent to g(0). Fix a local coordinate domain U ⊂⊂Mwith coordinates xi, and use T˜ to denote the local components in U of aglobal tensor T onM . Recall that if Γ˜(t) are the Christoffel symbols of g˜(t),then ∂∂t Γ˜(t) are the components of a global tensor g−1 ∗∇Rc on M for eacht. Combining the uniform equivalence of g(t) shown above with (3.27) fork = 1 we obtain∂∂tΓ˜(t) ≤ C1/t1+c1 .Thus Γ˜(t) remains uniformly bounded in U for all t. Now if we let ∂ denotepartial derivatives in the coordinates xi, then in U we have∂∂t∂g˜ = −2∂R˜c = −2(∇˜Rc + Γ˜ ∗ R˜c) ≤ C2/t1+c2 (3.28)by the boundedness of Γ˜(t) and again the uniform equivalence of g(t) shownabove and (3.27) for k = 0, 1. From this we conclude that ∂g˜(t) convergesto a bounded limit on U as t→∞.By applying a partial derivative ∂ to (3.28), we may express ∂2∂t2∂g˜ interms of ∇˜mRc for m = 0, 1, 2 and ∂l∂tlΓ˜(t) for l = 0, 1 and we can similarlyshow these terms are bounded by C3/t1+c3 from which we conclude that∂2g˜(t) converges to a bounded limit in U as t→∞. Continuing inductivelyin this way, we can show that ∂mg˜(t) converges to a bounded limit in U ast → ∞ for each m = 3, 4, .... As U was an arbitrary coordinate chart, thisestablishes that g(t) converges in C∞loc(M) as t→∞ to a smooth limit g(∞)which is equivalent to g(0) (and thus is necessarily complete) and must alsobe flat by (3.27).We have so far shown that (M, g(∞)) is a complete flat manifold. In par-ticular, its universal cover is Rn with the Euclidean metric gE and therefore(M, g(∞)) is isometric to(Rn, gE)/Γ72where Γ is a discrete group of isometries that acts freely on Rn. On theother hand, we know that g(∞) satisfies the Sobolev inequalitySob(M, g(∞)) ≤ [A0, 0]because the above inequality holds for all t > 0 and g(t)→ g(∞) uniformlyin L∞(M). Then by Lemma 2.5.1, we conclude that (M, g(∞)) has a positiveasymptotic volume ratio which implies that the group of isometries Γ istrivial. This completes the proof.This completes the proof of Theorem A.3.3 Unbounded curvature initial dataIn this section we will prove Theorem B, which follows from Theorem Aand a diagonal argument using Hamilton’s Compactness Theorem for Ricciflows.First, recall the statement of the theorem.Theorem. With n ≥ 3 and δ(n), C0(n) as in Theorem A, the follow-ing holds. Suppose that (Mn, g) is a complete Riemannian manifold (notnecessarily bounded curvature) which is the pointed smooth local limit ofa sequence of complete bounded curvature Riemannian manifolds (Mi, gi)(i = 1, 2, . . . ) such that for each i ∈ N,Sob(Mi, gi) ≤ [Cgi , 0]andsupi∈N(∫M|Rm|n2 dVgi) 2n≤ δ(n)supi∈NCgi.Then there exists a complete solution g(t) to the Ricci flow on M × [0,∞)with g(0) = g that satisfies the curvature estimatesupx∈M|Rm|(x, t) ≤ C0(n)tfor all t ∈ (0,∞). Moreover if supi∈N ∥|Rm|∥Lp(gi) < ∞ for some p < n2 ,then g(t) converges locally smoothly to a complete flat limit metric g(∞) onMn with Euclidean volume growth, and thus Mn is diffeomorphic to Rn.73Proof. Let (Mi, gi) and (M, g) be as in the theorem. Thus there is an ex-haustion {Ui}∞i=1 of M by nested open sets, and maps ϕi : Ui →Mi that arediffeomorphisms onto their images, such that we have the convergenceϕ∗i (gi)C∞loc(M)−−−−−→ g.The hypothesis thatsupi∈N(∫M|Rm|n2 dVgi) 2n≤ δ(n)supi∈NCgiimplies either that (a) |Rm|gi ≡ 0 for all i ∈ N or else (b) that supi∈NCgi <∞. In the former case, we must necessarily have that |Rm|g ≡ 0 sincegi → g locally smoothly and the result follows. For the latter case, weknow by Theorem A that for each i ∈ N, there exists a long-time completeRicci flow gi(t) with gi(0) = gi. Moreover, the {gi(t)} uniformly satisfy thecurvature estimatesupx∈M|Rm|gi(t) ≤C0(n)t(3.29)for all t ∈ (0,∞). Now to apply Hamilton’s Compactness to obtain a limitsolution on M from the sequence {gi(t)}, we need to obtain a positive lowerbound on the injectivity radius inj(gi(t0), xi) ≥ C where xi = ϕi(x) for somefixed point x ∈M and C, t0 > 0 and are independent of i.By (3.29), there exists r0(n) > 0 small enough so thatsupi∈Nsupx∈Bgi(1)(xi,r0)|Rm|gi(1)(x) ≤ r−20 . (3.30)Moreover, by the proof of Theorem, A (specifically Corollary 3.2.3), eachgi(1) satisfies the L2-Sobolev inequality Sob(Mi, gi(1)) ≤ [2L(n) supi∈NCgi , 0]and thus also (by Lemma 2.5.1) a uniform lower bound on volume ratiosVolgi(1)Bgi(1)(xi, r0) ≥ v0rn0 (3.31)where v0 > 0 depends on n and supi∈NCgi . Then by (3.30), (3.31) andresults of Cheeger-Gromov-Taylor [10], there holds the uniform estimateinjgi(1)(xi) ≥ c(n, r0, v0) > 0which is independent from i ∈ N, as was desired.74Now armed with the above, we can apply Hamilton’s Compactness The-orem [23], coupled with a diagonal argument, to obtain the convergenceϕ∗ik(gik(t))C∞loc(M×(0,∞))−−−−−−−−−−→ g(t)for some subsequence {ik}, where g(t) satisfies - by virtue of the smoothconvergence - the same curvature estimate (3.29).To obtain the convergence of g(t) to g as t → 0 we use (3.29) to apply[11, Corollary 3.2] which states thatsupBgi(0)(xi,rx,i2) |Rm|gi(t) ≤ Cr−2x,i .Here C = C(n) > 0 is a dimensional constant and rx,i > 0 is such thatsupBgi(0)(xi,rx,i)|Rm|gi(0) ≤ r−2x,i .Moreover for any fixed point x ∈M , the infimumrx := infi∈Nrx,iis strictly positive because of the local smooth convergence gi(0) → g(0).Therefore we have the locally uniform curvature estimatesup{|Rm|gi(t)(y) : i ∈ N, t ∈ [0, 1], y ∈ Bgi(0)(xi,rx2)}≤ C(n)r−2x .So because the flows {gi(t)} converge for all positive times to g(t) the con-vergence must occur at t = 0 as well. In other words, we have the smoothlocal convergence up to t = 0 asϕ∗ik(gik(t))C∞loc(M×[0,∞))−−−−−−−−−→ g(t).The completeness of the limiting flow g(t) follows from the Shrinking BallsLemma [41, Corollary 3.3] because the C0/t curvature bound (3.29) descendsto the limit flow g(t), and g(0) = g is assumed to be complete.The convergence statement follows from the proof of Proposition 3.2.11because the estimates obtained there only depend on n, p, Cgi and ∥|Rm|∥Lpgiwhich are all assumed to be uniform with respect to i ∈ N. Thus theestimates (3.26) and (3.27) pass down to the limit g(t). This gives theuniform convergence of g(t) to a flat metric g(∞) on M . The completenessof the limiting metric, and subsequently the topological statement aboutM ,are then identical to the proof of Proposition 3.2.11.753.4 An explicit bound for L(n)In this section, we will compute an explicit bound for the constant L(n)that appears in Theorem 2.4.1, and thus also an explicit bound on theconstant δ(n) in Theorem A.In the proof of Theorem 2.4.1, we establish that for any t ∈ [0, T ] andany σ > 0, there holds the following log-Sobolev inequality:∫u2 log u2 dVt ≤ σ(∫|∇u|2 + R+4dVt)− n2log σ +n2(logCg + log n− 1) .(3.32)As described in Section 2.4, we can now fix a time t ∈ [0, T ] in this log-Sobolev inequality and from it obtain an L2-Sobolev inequality at time t. Wewill track through the proof presented by Ye [49, Appendix C] to calculatethe constants in this process. Although Ye restricts to compact manifolds,the proof is verbatim (with the same constants) for bounded domains Ω ⊂Mand Dirichlet boundary conditions (see [14, Appendix B]). As commentedon in Sections 2.3 and 2.4, working on bounded domains with boundary issufficient for our purposes.For fixed t ∈ [0, T ], we use (3.32) as the hypothesis for [49, Theorem 5.3]withβ(σ) = −n2log σ +n2(logCg + log n− 1) .The output of this theorem is that∥e−sHu∥L∞ ≤ eτ(s)∥u∥L2whereτ(s) =12s∫ s0β(σ) dσ= −n212s(s log s− s) + n4(logCg + log n− 1)= −n4log s+n4(logCg + log n)= −n4log s+n4log(nCg).76Then applying [49, Theorem 5.4] with c = (nCg)n4 , µ = n, and trackingthrough the constants in the proof, we obtain(∫u2nn−2 dVt)n−2n≤ K2Cp0Cp1(∫|∇u|2 + R+4u2 dVt).So we can take A = K2Cp0Cp1 whereK and Cpi are as follows. HereK is theconstant in the Marcinkiewicz Interpolation Theorem, which we will obtaina bound for later (see Theorem 3.4.2). We will take p0 ∈ (1, 2) variable (tobe optimized later), with p1, q0, q1 defined by1p0+1p1= 1, and qi =npin− pi .The constants Cpi are then given generally asCp =(2p(2n−p)n−p cp2n−pp Γ(12)−p) 1q,wherecp = Γ(12)−1 (2n2 c2) 1p 2pn− p.Calculating each of the terms involved in Cp, we have(2p(2n−p)n−p 2n2pp2n−p)n−pnp= 22n−pn 212 = 25n−2p2n ,and Γ(12)− p2n−pΓ(12)−pn−pnp=√π−1.Recall c = (nCg)n4 , so thatc2pp2n−pn−pnp = (nCg)n42pp2n−pn−pnp = (nCg)12 .ThereforeA = K2Cp0Cp1= K2π−125n−2p02n+5n−2p12n (nCg)(2p0n− p0) p0n(2p1n− p1) p1n= CgK2[nπ25(p0n− p0) p0n(p1n− p1) p1n].(3.33)77Now we need to compute a bound for the constant K which, as notedabove, comes from an application of the off-diagonal Marcinkiewicz Interpo-lation Theorem (specifically, we require C1−t0 Ct1 dependence in the strong-type estimate where C0, C1 are the weak-type constants). There are manycitations for this particular result. For instance, a proof in a more generalsetting is given in [21, Theorem 1.4.19] and the upper bound that is obtainedafter plugging in our constants is something like K ≈ 1000 ∗ 212n. Throughan argument communicated to the author by Espinoza [35], one can obtainmuch sharper bounds by combining the ideas in [20] and [21]. We duplicatethe argument necessary for this change in the proceeding subsection. Fornow, let us see how it is being applied.In our setting, we are applying Theorem 3.4.2 with our given pi, qi andwith t = 12 . Thus we haveK = 2(qq − q0 +qq1 − q) 1q.Plugging this back into (3.33), we haveA = Cg[2(qq − q0 +qq1 − q) 1q]2 [nπ25(p0n− p0) p0n(p1n− p1) p1n].Recall that we did all this with a variable p0 ∈ (1, 2). We can optimize thisbound by finding its infimum over this interval. Thus we takeL(n) = infp0∈(1,2)n25π[2(qq − q0 +qq1 − q) 1q]2(p0n− p0) p0n(p1n− p1) p1n .For any n ≥ 3, this infimum is achieved for some p0 ∈ (1, 2). We cancalculate numerical bounds for L(n) using a computer:L(n) <{646 n = 3109n n ≥ 4 (3.34)Remark 3.4.1. In [49, Theorem 5.4], Ye gives precise values of p0, p1 ofp0 =n+ 2nand p1 =n+ 22.783.4.1 Modified MarcinkiewiczTheorem 3.4.2 ([35] Off-Diagonal Marcinkiewicz Interpolation Theoremwith Improved Constant). Suppose that (X,µ) and (Y, ν) are σ-finite mea-sure spaces and that p0, p1, q0, q1 ∈ [1,∞] such that p0 < p1, p0 < q0, p1 < q1,q0 < q1, and define for t ∈ (0, 1), pt and qt by1pt=1− tp0+tp1,1qt=1− tq0+tq1. (3.35)If T : Lp0(X,µ)+Lp1(X,µ)→ L0(ν) is a sub-additive operator which is weaktype (pi, qi) with constant Ci, then T is strong type (pt, qt) with constantCt = 2(qtqt − q0 +qtq1 − qt) 1qtC1−t0 Ct1.Proof (Duplicated): Fix some t ∈ (0, 1) and, for ease of notation, denotep = pt and q = qt. Let f ∈ Lp(X,µ) be normalized so that ∥f∥p = 1. Wewill show that∥Tf∥q ≤ 2(qq − q0 +qq1 − q) 1qC1−t0 Ct1. (3.36)Equations (3.35) imply thatp0(q − q0)q0(p− p0) =1p(1q0− 1q)1q(1p0− 1p) = 1p(1q − 1q1 )1q(1p − 1p1) = p1(q1 − q)q1(p1 − p) . (3.37)Let us denote by σ this common value. For arbitrary α > 0 and a selectedϵ > 0 (to be determined later) we havef = fχ{|f |>ασϵ} + fχ{|f |≤ασϵ} =: g + h.Notably, g ∈ Lp0(µ) and h ∈ Lp1(µ). By the (p0, q0) and (p1, q1) weak-typeestimates, we haveν(|Tf | > α) ≤ ν(|Tg| > α/2) + ν(|Th| > α/2)≤(2C0α)q0 (∫{|f |>ασϵ}|f |p0 dµ) q0p0+(2C1α)q1 (∫{|f |≤ασϵ}|f |p1 dµ) q1p1.79Consequently, there holds∥Tf∥qq = q∫ ∞0αq−1ν(|Tf | > α) dα≤ q(2C0)q0∫ ∞0αq−q0−1(∫{|f |>ασϵ}|f |p0 dµ) q0p0 dα+ q(2C1)q1∫ ∞0αq−q1−1(∫{|f |≤ασϵ}|f |p1 dµ) q1p1 dα.The assumptions of pj < qj and the σ-finiteness of the spaces involved allowsfor us to apply Minkowski’s inequality to each of the two integrals on theright-hand side. This gives∫ ∞0αq−q0−1(∫{|f |>ασϵ}|f |p0 dµ) q0p0dαp0q0≤∫X|f |p0∫ |f |1/σϵ1/σ0αq−q0−1 dαp0q0dµ=ϵ− q−q0σp0q0(q − q0)p0q0∫X|f |p0 |f |q−q0σp0q0 dµ=ϵ−(p−p0)(q − q0)p0q0∫X|f |p dµwhere the last equality follows from (3.37). Similarly∫ ∞0αq−q1−1(∫{|f |≤ασϵ}|f |p1 dµ) q1p1dαp1q1≤ ϵp1−p(q1 − q)p1q1∫X|f |p dµ.Putting things together, we obtain that for all f ∈ Lp(µ) with ∥f∥p = 1,there holds∥Tf∥qq ≤ q((2C0)q0ϵ− q0p0(p−p0)q − q0 +(2C1)q1ϵq1p1(p1−p)q1 − q).Now we choose ϵ > 0 so that(2C0)q0ϵ− q0p0(p−p0) = (2C1)q1ϵq1p1(p1−p) (3.38)80which subsequently yields∥Tf∥qq ≤(qq − q0 +qq1 − q)(2C1)q1ϵq1p1(p1−p). (3.39)Equation (3.38), along with (3.37), implies that(2C0)q0(2C1)q1= ϵq1p1(p1−p)+ q0p0 (p−p0) = ϵq1−qσ+q−q0σ = ϵq1−q0σ .Using this value of ϵ, our definition of σ from (3.37), and our assumption onpi, qi as in (3.35) yields(2C1)q1ϵq1p1(p1−p) = (2C1)q1((2C0)q0(2C1)q1)σ q1(p1−p)p1(q1−q0)= (2C1)q1((2C0)q0(2C1)q1) q1−qq1−q0= (2C0)q1q− 1q11q0− 1q1 (2C1)q1−q1 q1−qq1−q0= (2C0)q(1−t)(2C1)qq1(1q− (1−t)q0)= 2qCq(1−t)0 Cqt1 .Putting this back into (3.39) and taking q’th roots on both sides gives (3.36)and completes the proof.3.5 Wrinkled ballsIn this section, we justify Remark 1.2.4 by constructing a family of ex-amples of Riemannian metrics on Rn with unbounded curvature, while theglobal curvature concentration and global L2-Sobolev inequality are arbi-trarily close to that of flat Euclidean space. More precisely, we prove thefollowing.Proposition 3.5.1. Given any ϵ > 0, there exists an unbounded curvatureRiemannian metric g on Rn such that(1 + ϵ)−1gE ≤ g ≤ (1 + ϵ)gE (3.40)and ∫Rn|Rm|n2 dVg < ϵ81Before proceeding with the proof of Proposition 3.5.1, we point out thatthe condition that the metric g is C0-close to the Euclidean metric gE isactually a stronger hypothesis than requiring that the L2-Sobolev constantof g is arbitrarily close to that of Euclidean space. This is because eachterm in an L2-Sobolev inequality (namely, the volume form dVg and thenorm of the gradient |∇gu|2g) are zeroth order and depend continuously onthe metric. All that is to say that we could replace condition (3.40) abovewith the weaker conditionSob(Rn, g) ≤ [CgE + ϵ, 0]if desired.We begin with an intermediate lemma where we compactly modify thestandard flat Euclidean metric on a ball, by adding a small ‘wrinkle’ in theball. This allows us to alter the norm of the curvature drastically whilekeeping the integral curvature small.Lemma 3.5.2 (Wrinkled Ball). For any C ≥ 1 and ϵ > 0, there exists asmooth metric gC,ϵ on Ω = BgE (0, 2) which satisfies• gC,ϵ = gE on BgE (0, 1/2) and Ω \BgE (0, 1),• (1 + ϵ)−1gE ≤ gC,ϵ ≤ (1 + ϵ)gE,• supΩ |Rm|g > C, and•∫Ω |Rm|n2 dVg < ϵ.Proof. Fix a C ≥ 1 and ϵ > 0. We start with defining a smooth bumpfunction ψ : R→ [0, 1] which is supported on [0, 1] and nonzero. Define fora, b > 0 the functionψa,b(x) = aψ(xb).First we set b > 0 to beb = min{ϵVolgE (Sn−1)(2C)n2,12}(3.41)where Sn−1 is the (n− 1)-Euclidean unit sphere. We will also write ◦gn−1 forthe standard round metric on Sn−1. For a > 0, we write define a metric ha,bon Sn−1 × (0, 2) byha,b = dr2 + (r − ψa,b(r − 1 + b))2 ◦gn−1.82Note that this metric only deviates from gE on BgE (0, 2) by the addition ofthe ψa,b term, which effectively adds a ‘wrinkle’ in an otherwise standardEuclidean ball, as visualized in Figure 3.1 below.Figure 3.1: A wrinkle is added to the interior of Euclidean ballNow choose a > 0 to be small enough so that each of the followingconditions is satisfied:• ψa,b ≤ 18 ,• (1 + ϵ)−1gE ≤ ha,b ≤ (1 + ϵ)gE , and• supΩ |Rm|ha,b ≤ 1.Note the first property is true because we’ve required b ≤ 1/2 in (3.41) aswell as how the ψ function contributes to the metric. The latter conditionscan be achieved by taking a small enough because ∥ha,b− gE∥C2gE (Ω) → 0 asa→ 0. Now with this fixed set of values for a, b > 0, define for ρ ∈ (0, 1] themetric gρ = hρa,ρb. A few important properties by the construction are• gρ = gE on BgE (0, 1/2) and Ω \BgE (0, 1),• (1 + ϵ)−1gE ≤ ha,b ≤ (1 + ϵ)gE , and• supΩ |Rm|gρ →∞ as ρ→ 0.Note that the second property is true because the addition of the small ρonly decreases ∥hρa,ρb−gE∥C0gE (Ω). The latter property is true by the second83property (C0-closeness to gE) as well as the fact that(Rmgρ)rrjj =∇ ∂∂rgrad(r − ψρa,ρb(r − 1 + ρb))r − ψρa,ρb(r − 1 + ρb)=d2dr2(r − ψρa,ρb(r − 1 + ρb))r − ψρa,ρb(r − 1 + ρb)=−1ρψ′′a,br − ψa,b .Since a, b > 0 are fixed, this quantity is unbounded as ρ → 0 on the regionin which gρ is not flat. Since ρ 7→ supΩ |Rm|gρ is continuous, we may chooseρ > 0 small enough so thatC < supΩ|Rm|gρ < 2C.Set g = gρ and notice that∫Ω|Rm|n2 dVg < (2C)n2VolgE (Sn−1)ρb < ϵby our condition on b in (3.41).Proof of Proposition 3.5.1. Choose any sequence of large numbers Cj →∞,and writeϵj =ϵ2j.Let gCj ,ϵj be the metric constructed in Lemma 3.5.2, albeit centered at thepoint xj := (4j, 0, 0, . . . , 0) instead of at the origin. Define the metric g onRn byg(x) ={gj(x) if x ∈ BgE (xj , 2)gE(x) otherwise.The resulting metric is smooth because so is each gj , there is no overlap,and because gj = gE on a neighborhood of the boundary of BgE (xj , 2). Theclaims on the curvature, C0-closeness to gE , and curvature concentration allfollow immediately by the properties in Lemma 3.5.2.84Chapter 4A Gap TheoremIn this chapter, we will prove Theorem C, which is the capstone resultof this dissertation. Recall, its statement is as follows.Theorem. For n ≥ 4, there exists a dimensional constant ϵ(n) ≥ (10n)−10n3such that the following holds. Let (Mn, g) be any complete Riemannian man-ifold (not necessarily bounded curvature) with nonnegative Ricci curvatureRcg ≥ 0. Suppose that (Mn, g) has positive asymptotic volume ratioAVR(g) ≥ v > 0,and curvature concentration satisfying∫M|Rm|n2 dVg ≤ ϵ(n)v.Then (Mn, g) is isometric to (Rn, gE).The organization of this chapter is as follows:• In Section 4.1, we prove that Theorem C follows (quite readily, at that)from the long-time Ricci flow result, Theorem D. Thus, the latter isour primary objective.• In Section 4.2 we discuss how distances evolve under the Ricci flow,proving in particular, a version of the so-called “Expanding BallsLemma” where the time interval depends only on the dimension, α/tcurvature bound, and lower bound on scalar curvature.• In Section 4.3, we give a modified version of Lemma 3.2.1, which isan evolution inequality for the local curvature concentration in thespecial case of when the cutoff function ϕ is a subsolution of the heatequation.85• In Section 4.4, we prove a short-time Ricci flow existence result forbounded curvature metrics with the property that the bound on thelocal curvature concentration is preserved (up to a dimensional con-stant).• Finally in Section 4.5, we prove Theorem D by approximating blow-downs of (M, g) by bounded curvature metrics, from which our short-time bounded curvature Ricci flow existence result is readily applied.This proves the existence of a long-time Ricci flow solution emanatingfrom g with the desired properties given in (1.9), and thus also withthe faster than 1/t curvature decay. As mentioned above, this resultis sufficient to prove our gap theorem result, Theorem C.4.1 Proof of Theorem C from Theorem DHere we provide a brief proof of our main result Theorem C from ourauxiliary long-time Ricci flow existence result, Theorem D. The method ofproof will be familiar to experts. First, the scale-invariant properties of thehypotheses of Theorem D allow us to apply Hamilton’s Compactness The-orem to a sequence of blow-down flows to extract a converging subsequenceon (0, 1]. Then the faster than 1/t curvature decay and volume estimatestogether imply that this limiting flow must be isometric to the flat Eu-clidean flow on Rn. The rigidity of the original metric can subsequently beconcluded using distance distortion of Ricci flows. Here are the details.Proof of Theorem C (assuming Theorem D). Let g(t), t ∈ [0,∞) be thelong-time Ricci flow emanating from g given by Theorem D. Now takeany sequence of times ti → ∞ and define the parabolically rescaled flowsgi(t) =1tig(tit), each defined on [0, 1]. Note that the curvature bound (ii)and the injectivity radius bound (iii) survive this rescaling, and so by Hamil-ton’s Compactness Theorem [23] and a diagonal argument, the (M, gi(t), x0)subconverges (in the pointed Cheeger-Gromov sense) to some limiting com-plete flow (M˜, g˜(t), x∞), t ∈ (0, 1] which, by nature of the faster than 1/tcurvature decay (v), must satisfy|Rm|g˜(t) ≡ 0 for each t ∈ (0, 1].Furthermore, for each t ∈ (0, 1], the metric g˜(t) must have a positive asymp-totic volume ratio because (iv) passes to the limit as well. Therefore, for86each t ∈ (0, 1], we must have that (M˜, g˜(t)) is isometric to (Rn, gE). Thusif we fix any δ > 0 and t ∈ (0, 1] small enough (depending on δ and n), wecan say that for all sufficiently large i ∈ N, there holdsVolgi(0)Bgi(0)(x0, 1) ≥ Volgi(t)Bgi(0)(x0, 1)≥ Volgi(t)Bgi(t)(x0, 1− β√t)≥ (1− δ)ωn(1− β√t)n≥ (1− 2δ)ωn,where we have used the nonnegative scalar curvature of the flow (i) in thefirst inequality, the Shrinking Balls Lemma [41, Corollary 3.3] (or referenceLemma 4.2.1 in this work) in the second (here β = β(n) is a dimensional con-stant), the convergence gi(t)→ gE in the third, and our smallness conditionon t in the fourth. Scaling back tells us thatlimr→∞VolgBg(x0, r)rn≥ (1− 2δ)ωn.Since δ > 0 was arbitrary, we can use our assumption of Rcg ≥ 0 again toin fact see thatlimr→∞VolgBg(x0, r)rn= ωn.Therefore (Mn, g) is isometric to (Rn, gE) by the rigidity statement in theBishop-Gromov Volume Comparison Theorem.4.2 Evolution of distanceIn this section, we give two results which describe how the distancefunction behaves under the Ricci flow. In particular, these results will allowus to conclude that the radius of geodesic balls stay uniformly controlledboth forward and backward in time. The first is the well-known “ShrinkingBalls Lemma” [41, Corollary 3.3] which controls the rate at which distancescan contract.Lemma 4.2.1 (Shrinking Balls [41]). For any n ∈ N and β(n) = 8√2/3(n−1) ≥ 1, the following holds. Suppose (Mn, g(t)), t ∈ [0, T ] is a (not neces-sarily complete) n-dimensional Ricci flow such that Bg(0)(x0, r) ⊂⊂M and|Rm|g(t) ≤ c0/t on Bg(0)(x0, r)∩Bg(t)(x0, r− β√c0t) for each t ∈ (0, T ] andsome c0 > 0. Then for any t ∈ [0, T ], there holdsBg(t)(x0, r − β√c0t) ⊂ Bg(0)(x0, r).87Obtaining control in the other direction, i.e., the rate at which distancescan expand, is historically more challenging. If it is known that the flowenjoys a lower Ricci bound for all times, then Simon & Topping have provedan analogous “Expanding Balls Lemma” [41, Lemma 3.1]. The issue ishowever that lower Ricci curvature bounds are not respected under Ricciflow when n ≥ 4. Therefore it is desirable to replace the lower Ricci boundassumption with a lower scalar curvature bound (which is preserved) alongwith some volume estimates (see [24, Lemma 3.7] and [29, Lemma 2.2] forsuch results of this flavour). The improvement that we offer here is uniformcontrol on the rate of the expansion depending only on the quotient ofvolume ratios, albeit for a short amount of time. Importantly, the timeinterval does not depend on the given volume bounds.Lemma 4.2.2 (Expanding Balls). For any n ∈ N and any α, σ > 0, thereexists T̂ (n, α, σ) > 0 such that the following holds. Let (Mn, g(t)), t ∈ [0, T ]be a complete bounded curvature Ricci flow such that for some x0 ∈ M ,b0 > 0, and c0 > 0 there holds:(a) infx∈MRg(0)(x) ≥ −σ,(b) supx∈M|Rm|g(t)(x) ≤ αt−1 for all t ∈ (0, T ],(c) Volg(0)Bg(0)(x0, 2) ≤ b0, and(d) Volg(t)Bg(t)(x, r) ≥ c0rn whenever x ∈ Bg(0)(x0, 1), r ∈ (0, 12 ], andt ∈ (0, T ].Then for all t ∈ [0,min{T, T̂}], we have Bg(0)(x0, 1) ⊂ Bg(t)(x0, 2n+2b0c−10 ).Proof. We begin by setting our maximal time interval:T̂ = min{log 2σ,14β2α}.Here β = β(n) is from Lemma 4.2.1. Fix τ ∈ [0,min{T, T̂}] and defineR = inf{r > 0 : Bg(0)(x0, 1) ⊂ Bg(τ)(x0, r)}.We will show that R ≤ 2n+2b0c−10 which will prove the result.By the definition of R, there exists y ∈ M such that dg(0)(x0, y) = 1and dg(τ)(x0, y) = R. Let γ : [0, 1] → M be a minimizing unit-speed g(0)-geodesic with γ(0) = x0 and γ(1) = y. Notice that because T̂ ≤ 14β2α , we88have that Bg(τ)(x, 1/2) ⊂ Bg(0)(x, 1) for all x ∈ M by Lemma 4.2.1. Thusby taking z1 = x0, we can inductively choose some set of points {zj}kj=1 ⊂Bg(0)(x0, 1) such that {Bg(τ)(zj , 12)}kj=1 are mutually disjoint andk⊔j=1Bg(τ)(zj , 1/2) ⊂ Bg(0)(x0, 2) and Bg(0)(x0, 1) ⊂k⋃j=1Bg(τ)(zj , 1).Then since γ ⊂ Bg(0)(x0, 1), we can obtain a lower bound on dg(τ)(x0, y) byestimating the length of γ with respect to the metric g(τ) asR = dg(τ)(x0, y) ≤ Lengthg(τ)(γ) ≤ 2k. (4.1)Figure 4.1 below provides an illustration to help visualize this inequality.The circles are g(τ)-unit balls, which cover Bg(0)(x0, 1) and thus γ as well.Figure 4.1: Approximating Lengthg(τ)(γ) from unit g(τ)-ballsBy our assumptions on the bounds of volumes and the assumed disjoint-ness of {Bg(τ)(zj , 12)}kj=1, we can obtain an upper bound on k:kc0(1/2)n ≤k∑j=1Volg(τ)Bg(τ)(zj , 1/2)≤ Volg(τ)(Bg(0)(x0, 2))≤ 2Volg(0)(Bg(0)(x0, 2)) ≤ 2b0.(4.2)In the third inequality, we have used the evolution of the volume form∂tdVg(t) = −Rg(t)dVg(t), the lower scalar bound Rg(0) ≥ −σ (which is pre-served along complete bounded curvature flows), and the condition on theexistence time σT̂ ≤ log 2. By (4.1) and (4.2), we have the desired boundR ≤ 2n+2b0c−10 .89Remark 4.2.3. Because local lower bounds on the scalar curvature Rg(0)are semi-preserved along all Ricci flows with α/t curvature control [11], itis clear that Lemma 4.2.2 can be localized, i.e., for some compact regioncontained within a possibly incomplete and/or unbounded curvature flow,but the stated version will suffice for our purposes.4.3 Evolution of curvature concentrationIn this section, we prove a lemma demonstrating how the local curvatureconcentration evolves under the Ricci flow. This result differs from Lemma3.2.1 in the sense that here our choice of cutoff function ϕ is time-dependent;in fact, it is a subsolution to the heat equation. This modification paysdividends later in the proof of Theorem 4.4.2. The proof is essentially thatof [4, Lemma 2.1], but we provide a brief sketch to demonstrate explicitestimates on the coefficients.Lemma 4.3.1. Suppose (M, g(t)), n ≥ 3, t ∈ [0, T ] is a complete (possiblyunbounded curvature) Ricci flow and ϕ : M × [0, T ] → R is a smooth non-negative function with compact support. If □ϕ ≤ 0 (here □ = ∂t −∆g(t) isthe heat operator), then for any β > 0 there holdsddt∫ϕ2(|Rm|2 + β)n4 dVt≤ −13∫|∇(ϕ(|Rm|2 + β)n8 )|2 dVt+ 10n∫|∇ϕ|2(|Rm|2 + β)n4 dVt+ 5n∫ϕ2(|Rm|2 + β)n4+ 12 dVt.Proof. For convenience, we will denote α = n4 . We begin by differentiatingunder the integral, integrating by parts (since suppϕ is compact), using theevolution of volume ∂t dVt = −RdVt, and |R| ≤ n|Rm| (see (1.19)) which90givesddt∫ϕ2(|Rm|2 + β)α dVt≤∫∂∂t(ϕ2(|Rm|2 + β)α) dVt + ∫ ϕ2|R|(|Rm|2 + β)α+ 12 dVt≤∫□(ϕ2(|Rm|2 + β)α) dVt + n ∫ ϕ2(|Rm|2 + β)α+ 12 dVt.For the □ term, we calculate∫□(ϕ2(|Rm|2 + β)α) dVt=∫2ϕ□ϕ(|Rm|2 + β)α dVt −∫2|∇ϕ|2(|Rm|2 + β)α dVt+∫αϕ2(|Rm|2 + β)α−1□|Rm|2 dVt−∫4α(α− 1)ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt−∫8αϕ⟨∇ϕ,∇|Rm|⟩|Rm|(|Rm|2 + β)α−1 dVt= I+ II+ III+ IV+V.Clearly II ≤ 0, and since □ϕ ≤ 0, we have I ≤ 0 as well. Also, by theevolution of the curvature tensor and the Kato inequality |∇|T || ≤ |∇T |, wehaveIII ≤ −2α∫ϕ2(|Rm|2 + β)α−1|∇Rm|2 dVt + 16α∫ϕ2(|Rm|2 + β)α+ 12 dVt≤ −2α∫ϕ2(|Rm|2 + β)α−1|∇|Rm||2 dVt + 16α∫ϕ2(|Rm|2 + β)α+ 12 dVt≤ −2α∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt+ 16α∫ϕ2(|Rm|2 + β)α+ 12 dVt = IIIa+ IIIb.91Applying Cauchy-Schwarz and Young’s inequality givesV ≤∫8αϕ|∇ϕ||∇|Rm|| |Rm|(|Rm|2 + β)α−1 dVt≤ 8α(∫|∇ϕ|2(|Rm|2 + β)α dVt) 12×(∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt) 12≤ 4αϵ∫|∇ϕ|2(|Rm|2 + β)α dVt+ 4αϵ∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt = Va+Vb.Now gather up the terms IV, IIIa, Vb, and set ϵ = 18 so that the coefficientis c(α) := 32α + 4α(α − 1), which satisfies −c(α) ∈ [23α2, 4α2]. Then usingCauchy-Schwarz again givesc(α)∫ϕ2(|Rm|2 + β)α−2|Rm|2|∇|Rm||2 dVt= c(α)1α2∫ϕ2|∇(|Rm|2 + β)α2 |2 dVt≤ c(α) 1α2(12∫|∇ϕ(|Rm|2 + β)α2 |2 dVt −∫|∇ϕ|2(|Rm|2 + β)α dVt)≤ −13∫|∇ϕ(|Rm|2 + β)α2 |2 dVt + 4∫|∇ϕ|2(|Rm|2 + β)α dVt.The result follows by gathering like terms and simplifying.4.4 A bounded curvature localized resultAs part of the proof of Theorem D, we will require a short-time Ricciflow existence result for bounded curvature metrics with the properties thatthe flow enjoys 1/t curvature control and preservation of the local curvatureconcentration (see Theorem 4.4.2). This result is similar to [5, Theorem 3.1],which in turn is only a slight generalization of [4, Theorem 1.2]. There is akey difference between our Theorem 4.4.2 and these other results however,and that is that our hypothesis are permitted to vary, as long as the constantslie within a given range. This allowance is what allows us to leverage the92equivalence between volume ratios and Sobolev inequalities in spaces withnonnegative Ricci curvature later on. Because of this key difference, we willpresent a full proof of Theorem 4.4.2 even though much of the process issimilar to the synonymous results in [4, 5].Before stating that however, we first prove an elementary lemma whichstates that the local curvature concentration can only grow at most at someuniform rate under bounded curvature flows.Lemma 4.4.1. Let (Mn, g(t)), t ∈ [0, T ], n ≥ 3 be a complete boundedcurvature Ricci flow such thatsupx∈M∫Bg(0)(x,r1)|Rm|n2 dVg(0) ≤ σfor some r1 > 0, σ > 0. Then for any r2 < r1 and ϵ > 0, there exists η > 0such that for each t ∈ [0, η], there holdssupx∈M∫Bg(t)(x,r2)|Rm|n2 dVg(t) ≤ σ + ϵ.Proof. For the sake of brevity, we write K = sup |Rm(x, t)| + 1 where thesupremum is taken over all (x, t) ∈M× [0, T ]. In what follows, we will writeC to be some constant possibly depending on n, r1, r2, σ,K which may varyline-by-line. Let ψ : R→ R be some smooth decreasing cutoff function suchthat ψ ≡ 1 on (−∞, 0], ψ ≡ 0 on [1,∞) and ψ′ ≥ −2. Now for any x ∈M ,let ϕx :M → R be defined asϕx(y) = ψ(2r1 − r2dg(0)(x, y)−r1 + r2r1 − r2).Then suppϕx ⊂ Bg(0)(x, r1), ϕx ≡ 1 on Bg(0)(x, r1+r22 ), and |∇g(0)ϕx|2g(0) ≤C. By the fact that the flow has bounded curvature, we may assume thatη > 0 is small enough so that for all (x, t) ∈M × [0, η], we haveBg(t)(x, r2) ⊂ Bg(0)(x, r1+r22 ),|∇g(t)ϕx|2g(t) ≤ 2|∇g(0)ϕx|2g(0), andVolg(t)Bg(0)(x, r1) ≤ C.Now for any 0 < β ≤ K, we apply Lemma 3.2.1 to see thatddt∫ϕ2(|Rm|2 + β)n4 dVt≤ C∫ϕ2(|Rm|2 + β)n4+ 12 dVt + C∫|∇ϕ|2(|Rm|2 + β)n4 dVt ≤ C.93Integrating in time then sending β → 0, we can apply our properties of ϕ toobtain ∫Bg(t)(x0,r2)|Rm|n2 dVg(t) ≤∫ϕ2|Rm|n2 dVg(t)≤∫ϕ2|Rm|n2 dVg(0) + Ct≤∫Bg(0)(x0,r1)|Rm|n2 dVg(0) + Ct.The result follows by restricting η ≤ ϵ/C.Theorem 4.4.2. For all n ≥ 3, λ ≥ 1, v0 > 0, there exists C(n) ≥ 1,c(n) > 0, ι(n, λ, v0), δ(n, λ) > 0, Λ(n, λ) > 1, and T˜ (n, λ, v0) > 0 such thatthe following holds. Suppose that (Mn, g(t)), t ∈ [0, T ] is a complete boundedcurvature Ricci flow and the initial metric g(0) satisfies the following for allx ∈M :(a) Rcg(0)(x) ≥ −1;(b) Volg(0)Bg(0)(x, 2r) ≤ 22n+1 ·Volg(0)Bg(0)(x, r) ≤ 22n+2ωnrn for all r ≤4; and(c)∫Bg(0)(x,4)|Rm|n2 dVg(0) ≤ δ(n, λ)v0.Moreover for each x ∈M and r ∈ (0, 1], assume thatvxr :=Volg(0)Bg(0)(x, r)rn≥ v0and thatSob(Bg(0)(x, 4r), g(0)) ≤[λv− 2nxr , 0]. (4.3)Then the flow g(t) satisfies each of the following estimates uniformly for all(x, t) ∈M × [0,min{T˜ , T}]:(i) Rg(t)(x) ≥ −n;(ii) |Rm|g(t)(x) ≤ 1t ;(iii) injg(t)(x) ≥ ι(n, λ, v0)√t;(iv) Volg(t)Bg(t)(x, r) ≥ c(n)λ−n2 v0 rn for all r ∈ (0, 1];94(v)∫Bg(t)(x,1)|Rm|n2 dVg(t) ≤ Λ(n, λ)δ(n, λ)v0; and(vi) Sob(Bg(t)(x, r), g(t)) ≤[C(n)λv− 2nxr , 1]for all r ∈ (0, 1].In particular, the flow exists up until time T˜ .Remark 4.4.3. The lower Ricci bound assumption (a) can be weakenedto a lower bound on the scalar curvature Rg(0) ≥ −1 in the presence ofthe volume doubling assumption (b), at the expense of possibly shrinkingδ(n, λ). We choose to state the theorem like this because it is more naturalin our setting, and because this stronger hypothesis allows us to substitute alimiting contradiction argument for the Moser Iteration argument (see Step 2in the proof). This is beneficial to us because, not only is the Moser Iterationa very technical argument which is difficult to track precise constants, butthis gives us another opportunity to tie-in our Theorem A.Proof. Note that since lower scalar curvature bounds are preserved by com-plete bounded curvature flows, conclusion (i) is trivially satisfied (we haveincluded it in the theorem statement for clarity later). We begin the proofof the other properties by fixing our constants. First, we setT˜ := min{T1, C−14 ,116β−2,14C−13 ,log 210,r40104, r20T̂}. (4.4)Here T1 = T1(n, λ, v0) > 0 is some positive time coming from Step 2 of theproof, β(n) is from the Shrinking Balls Lemma 4.2.1, T̂ = T̂ (n, 1, 1) is fromthe Expanding Balls Lemma 4.2.2, and c1(n), C3(n), C4(n, λ), r0(n, λ) areconstants that will be defined within the proof. Also, we defineδ(n, λ) = Λ(n, λ)−1(1107n3λ)n2, (4.5)where Λ(n, λ) is defined in (4.21). We will see in Step 1 of the proof thatwe may takeC(n) = 1000n and c(n) = (2n+4 · 1000n)−n2 .Finally, recall that condition (iii) follows from the conclusions (ii) and (iv)by [10], so we will assume that the ι(n, λ, v0) > 0 is the constant given there.95Now we may proceed with the proof. Let T ′ be the maximal time in theinterval [0,min{T, T˜}] such that for all (x, t) ∈M × [0, T ′], there holds∫Bg(t)(x,1)|Rm|n2 dVg(t) ≤ Λ(n, λ)δ(n, λ)v0. (4.6)Note that because (M, g(t)), t ∈ [0, T ] is a bounded curvature flow whichsatisfies (c), Lemma 4.4.1 guarantees that T ′ > 0. In advance however, itappears as though T ′ depends on the curvature bound of the flow g(t), butwe will show that this is not the case.Our proof is broken down into three steps: (1) Show we have a uniformlocal Sobolev inequality, i.e., conclusion (vi), along the flow on the interval[0, T ′] assuming the time interval is possibly shortened so that a global α/tcurvature bound holds, which in particular implies conclusion (iv); (2) Showthat the assumption of the global α/t curvature bound assumption in Step 1is implied as long as T ′ ≤ T˜ (this gives conclusions (ii) and (iii)); (3) Derivea contradiction in the event that T ′ < min{T, T˜}.Step 1 (Sobolev inequality)Fix some r0(n, λ) ∈ (0, 1) to be defined later (in hindsight, we couldtake r0 = [23n+6(2n+41000nλ)n2 ]−1). Since (M, g(t)), t ∈ [0, T ] is a boundedcurvature flow, we may assume that there is some T1 > 0 such that|Rm|g(t) ≤3r0/10tfor all t ∈ (0, T1]. (4.7)In this step of the proof, we will restrict to the interval [0,min{T1, T ′}]. Wewill show in Step 2 that it is necessarily true that T1 can be taken to dependonly on the variables n, λ, v0.Fix some x ∈M and r ∈ [r0, 1]. By our smallness condition on δ(n, λ) in(4.5) and our assumptions (4.3) and (c), as well as the fact that vx,r ≥ v0,we may apply Lemma 2.2.1 (with a = 0) to obtainν(Bg(0)(x, 4r), g(0), 2) ≥ −n2log(C1(n)λv− 2nx,r ).Now by our curvature assumption (4.7), we apply the “almost-monotonicity”Lemma 2.1.2 (with the variables there being A = 3r10 , B =14 , D = 0, τ1 = 1)96to see that for each y ∈ Bg(0)(x, r) and t ∈ [0,min{T1, T ′}] there holdsν(Bg(t)(y, r), g(t), 1) ≥ ν(Bg(0)(y, 3r), g(0), 2)−(210(3r/10)2(1/4)2+ e−1)(eT ′10(3r/10)2(1/4)2 − 1)≥ ν(Bg(0)(x, 4r), g(0), 2)−n2log 2≥ −n2log(2C1(n)λv− 2nx,r ).Note that in the second inequality, we have used the fact that T ′ ≤ T˜ ≤ r40104so that (210(3r/10)2(1/4)2+ e−1)(eT ′10(3r/10)2(1/4)2 − 1)≤ 100r2(e50T ′r2 − 1) ≤ 104r−40 T˜ ≤ 1 <n2log 2.Since our smallness condition on δ in (4.5) in particular implies thatδ(n, λ) ≤ Λ(n, λ)−1(1250nλ)n2,then by Lemma 2.3.1 and Remark 2.4.2, we obtain (with C2(n) = 1000n)the uniform local Sobolev inequalitySob(Bg(t)(y, r), g(t)) ≤ [C2(n)λv−2nx,r , 1] (4.8)whenever y ∈ Bg(0)(x, r), which gives conclusion (vi). Note that by Lemma2.5.1, we can also say that{Volg(t)Bg(t)(y, ρ) ≥ c1(n)λ−n2 vx,rρn whenevery ∈ Bg(0)(x, r), ρ ∈ (0, r], t ∈ [0,min{T1, T ′}].(4.9)This gives conclusion (iv) since vx,r ≥ v0. Furthermore, note that c1(n) ≥(2n+4 ·C2(n))−n2 . We also point out that although the work in this step wasvalid for any r ∈ [r0, 1], we will only need to apply it with r = r0 and r = 1.This volume ratio estimate, together with the assumed curvature bound on(0,min{T1, T ′}] imply (by [10]) the injectivity radius boundinjg(t)(x) ≥ ι(n, λ, v0)√t for all x ∈M, t ∈ (0,min{T1, T ′}]. (4.10)This gives conclusion (iii).97Step 2 (Curvature bound)In this step of the proof, we will show that the flow g(t) satisfies thecurvature bound (4.7) as long as (4.6) holds on the time interval [0, T ′],assuming that T ′ ≤ T1 where T1 depends only on n, λ, v0. Just like theproof of [4, Proposition 2.1] and the proof of Theorem A, we could do this byapplying Moser’s iteration, which is well known to experts but is a somewhattechnical argument, especially in our setting where we must prove that thesmallness of∫ |Rm|n2 need only depend linearly on v. Instead, we will argueby contradiction where we employ a limiting contradiction argument withrespect to the existence time. Thus we may assume that for some n ≥ 3,λ ≥ 1, v0 > 0, there exists a sequence of times Ti → 0 and associatedbounded curvature Ricci flows (Mni , gi(t)), t ∈ [0, Ti] each satisfying thehypotheses of the theorem and such that∫Bgi(t)(x,1)|Rm|n2 dVgi(t) ≤ Λδv0for all (x, t) ∈ Mi × [0, Ti], but which do not enjoy a α/t curvature boundon the interval [0, Ti], where α = 3r0/10. Since each (Mi, gi(t)), t ∈ [0, Ti] isassumed to be a bounded curvature flow, we may assume that there is somefirst time ti ∈ [0, Ti] which fails our desired bound. Thus we may assumethat, for some xi ∈Mi, we have|Rm|gi(ti)(xi) ≥34αtiand that|Rm|gi(t) ≤αton Mi × (0, ti].Moreover by Step 1, we know that the Sobolev inequality (4.8) and volumeratio bound (4.9) hold with r = 1, ρ =√t, vx,1 = v0. Recall this in turngave the injectivity radius estimate (4.10) for each gi(t) on (0, ti].Now parabolically rescale the gi(t) so that ti ↗ 1, which is to say wedefine the sequence of complete flows g˜i(t) =1tigi(tit), each defined on [0, 1],98which satisfy Rcg˜i(0) ≥ −ti, Rg˜i(t) ≥ −nti,|Rm|g˜i(1)(xi) ≥ 34α,|Rm|g˜i(t) ≤ αt for all t ∈ (0, 1],∫Ωi|Rm|n2 dVg˜i(1) ≤ Λδv0,injg˜i(1)(xi) ≥ ι(n, λ, v0), andSob(Ωi, g˜(1)) ≤ [C2(n)λv−2n0 , ti].(4.11)Here we have denoted Ωi = Bg˜(1)(x, t− 12i ). Therefore we may apply Hamil-ton’s Compactness Theorem [23] and a diagonal argument to say that thesequence (Mi, g˜i(t), xi) subconverges (in the pointed Cheeger-Gromov sense)to some complete Ricci flow (N, g∞(t), x∞), t ∈ (0, 1], which by virtue ofthe properties (4.11), must satisfyRg∞(t) ≥ 0 for all t ∈ (0, 1],|Rm|g∞(1)(x∞) ≥ 34α,supx∈N |Rm|g∞(t)(x) ≤ αt for all t ∈ (0, 1],∫N |Rm|n2 dVg∞(1) ≤ Λδv0, andSob(N, g∞(1)) ≤ [C2(n)λv−2n0 , 0].Because C2(n) = 1000n, our requirement on δ(n, λ) in (4.5) implies thatδ(n, λ) ≤ Λ(n, λ)−1(δ(n)C2(n)λ)n2(4.12)where δ(n) ≥ 1104n2is the dimensional constant from Theorem A. Accordingto that theorem, we may therefore extend the flow g∞(t) to the interval(0,∞). Now by an identical blow-down argument as in the proof of TheoremC from Section 4.1, along with the faster than 1/t curvature decay, weconclude that for each t ∈ (0, 1], there holdslimr→∞Volg∞(t)Bg∞(t)(x∞, r)rn≥ ωn. (4.13)Now we have the following claim.Claim 4.4.4. For any A ≥ 1, there exists NA ∈ N such that wheneveri ≥ NA, there holdsinf0<t≤1µ(Bg˜i(0)(xi, 20A√t), g˜i(0), t) ≥ −A−2.99This claim will be sufficient to derive a contradiction because we maytake A = A(α/2, n) from [43, Theorem 4.5], which, in combination with(4.11), would yield the absurd inequality34α ≤ |Rm|g˜i(1)(xi) ≤α2.We will now prove the claim.Proof of Claim. For any given A ≥ 1 define η = A−2 and let ξ(η, n,A) > 0be the constant from [43, Lemma 4.10]. It will suffice to show that for all isufficiently large, we in fact have{Rcg˜i(0)(x) ≥ −ξ for all x ∈M ;Volg˜i(0)Bg˜i(0)(xi, r) ≥ (1− ξ)ωnrn for all r ∈ (0, ξ].(4.14)Indeed, for any t ∈ (0, 1], the metric 1t g˜i(0) will then also satisfy theseconditions which implies−A−2 ≤ µ(B 1tg˜i(0)(xi, 20A),1tg˜i(0), 1)= µ(Bg˜i(0)(xi, 20A√t), g˜i(0), t)where we have applied [43, Lemma 4.10] in the first inequality while thesecond follows from the scaling properties of the µ-functional.First let i be large enough to ensure that ti ≤ ξ, which shows the thelower Ricci bound condition. For the volume condition, we begin by definingξˆ > 0 by (1− ξˆ)4 = (1− ξ). Fix s ∈ (0, 1] small enough so thatRn ≥ (1− ξˆ)(R+ β√αs)n for all R ≥ ξ−1,where β = β(n) from Lemma 4.2.1. Now by the convergence g˜i(s)→ g∞(s)and the asymptotic volume estimate for g∞(s) as in (4.13), we can fix someR0 ≥ ξ−1 such thatVolg˜i(s)Bg˜i(s)(xi, R0) ≥ (1− ξˆ)ωnRn0 .We also assume that i is large enough so that ensti ≤ 1 + ξˆ. Finally bythe Bishop-Gromov Volume Comparison Theorem, we can say that for all ilarge enough and r ∈ (0, R0 + β√αs], there holdsVolg˜i(0)Bg˜i(0)(xi, r)rn≥ (1− ξˆ)Volg˜i(0)Bg˜i(0)(xi, R0 + β√αs)(R0 + β√αs)n.100Then for any r ∈ (0, ξ−1], these conditions on the largeness of i implyr−nVolg˜i(0)Bg˜i(0)(xi, r)≥ (1− ξˆ)(R0 + β√αs)−nVolg˜i(0)Bg˜i(0)(x0, R0 + β√αs)≥ (1− ξˆ)2(R0 + β√αs)−nVolg˜i(s)Bg˜i(0)(x0, R0 + β√αs)≥ (1− ξˆ)3R−n0 Volg˜i(s)Bg˜i(s)(x0, R0)≥ (1− ξˆ)4ωn = (1− ξ)ωn.This shows that condition (4.14) holds and completes the proof of the claim,and thus of Step 2 as well.Step 3 (Uniform time)We have already seen in Steps 1 and 2 that conclusions (i), (ii), (iii), (iv),and (vi) all follow as long as conclusion (v) holds on [0, T ′] if T ′ ≤ min{T, T˜}.We will assume that T ′ < min{T, T˜} then show that this contradicts themaximality of T ′ which will complete the proof of the theorem.Start by fixing some x0 ∈ M and let ϕ : M → R be defined as in [4].That is, ϕ(x, t) = e−10tφ(η(x, t)) where η(x, t) = dg(t)(x, x0) +C3(n)√t andφ : R → R is a smooth cutoff function with φ ≡ 1 on (−∞, 12 ], φ ≡ 0on [1,∞), φ′′ ≥ −10φ, 0 ≥ φ′ ≥ −10√φ. Moreover C3(n) is chosen largeenough so that □ϕ ≤ 0. With this choice of ϕ in Lemma 4.3.1, we haveddt∫Mϕ2(|Rm|2 + β)n4 dVt ≤ −13∫M|∇(ϕ(|Rm|2 + β)n8 )|2 dVt+ 5n∫Mϕ2(|Rm|2 + β)n4+ 12 dVt + 1000n∫suppϕ(|Rm|2 + β)n4 dVt.Now to the terms on the right-hand side, we apply our L2-Sobolev inequality(4.8) (here r = 1 and recall that vx,1 ≥ v0) to the first and Ho¨lder’s inequality101to the second to see thatddt∫Mϕ2(|Rm|2 + β)n4 dVt≤ − 13C2(n)λv− 2n0(∫Bg(t)(x,1)|ϕ(|Rm|2 + β)n8 | 2nn−2 dVt)n−2n+ 5n(∫suppϕ(|Rm|2 + β)n4 dVt) 2n×(∫Bg(t)(x,1)|ϕ(|Rm|2 + β)n8 | 2nn−2 dVt)n−2n+ 1001n∫suppϕ(|Rm|2 + β)n4 dVt.(4.15)Then integrate over [0, t] on both sides and send β → 0. Because suppϕ ⊂Bg(t)(x0, 1), our restriction on δ(n, λ) in (4.5) ensures that the sum of thetwo terms containing ∥ϕ2|Rm|n2 ∥Lnn−2 in (4.15) is nonpositive. Therefore wehave ∫Mϕ|Rm|n2 dVt ≤∫Mϕ|Rm|n2 dV0 + 1001nΛδv0t≤ δv0 + 1001nΛδv0t =: (1 + C4(n, λ)t)δv0.Now we multiply through by e10t and notice whenever t ≤ T ′ ≤ T˜ , theconditions in (4.4) imply e10t ≤ 2, C4(n, λ)t ≤ 1, and ϕ ≡ 1 on Bg(t)(x0, 1/4).Then for all such t ∈ [0, T ′], there holds∫Bg(t)(x0,14)|Rm|n2 dVt ≤ 4δv0. (4.16)Now fix some (y, t) ∈ M × (0, T ′] and let {xj} ⊂ Bg(t)(y, 3/2) be somesubset of points such that {Bg(t)(xj , 18)}Nj=1 are mutually disjoint, but sothatN⊔i=1Bg(t)(xi, 1/8) ⊂ Bg(t)(y, 7/4) (4.17)andBg(t)(y, 3/2) ⊂N⋃i=1Bg(t)(xi, 1/4). (4.18)102Then by the Shrinking Balls Lemma 4.2.1, since T ′ ≤ T˜ ≤ 116β2, we haveBg(t)(y, 7/4) ⊂ Bg(0)(y, 2). (4.19)We would like to be able to say that we have a similar reverse containment aswell. For this, we will be enlisting the help of our Expanding Balls Lemma4.2.2. Recall we have a α/t curvature bound by Step 2, a lower volumebound (4.9) by Step 1, and an assumed lower bound on scalar curvature (a).In summary, for all x ∈M , there holdsRg(0)(x) ≥ −n;|Rm|g(t)(x) ≤ αt for all t ∈ [0, T ′];Volg(0)Bg(0)(x, r0) =: vx,r0rn0 ; andVolg(t)Bg(t)(y, r) ≥ c1(n)λ−n2 vx,r0rnfor all y ∈ Bg(0)(x, r0), r ∈ (0, r0], t ∈ [0, T ′].(4.20)Now we parabolically rescale g(t) by definingg˜(t) = r−20 g(r20t)where r0(n, λ) =c1(n)23n+6λn2. Note that this rescaling has scaled distancesr0 → 1 and 18 → 23n+3λn2c1(n). Therefore, if we write T2 = r−20 T′, we can seefrom the properties in (4.20), and our volume doubling estimate (b), thatthe flow g˜(t), in particular, satisfiesRg˜(0)(x) ≥ −1;|Rm|g˜(t)(x) ≤ t−1 for all t ∈ [0, T2];Volg˜(0)Bg˜(0)(x, 2) ≤ 22n+1vx,r0 ; andVolg˜(t)Bg˜(t)(y, r) ≥ c1(n)λ−n2 vx,r0rnfor all y ∈ Bg˜(0)(x, 1), r ∈ (0, 1/2], t ∈ [0, T2].Then Lemma 4.2.2 implies that Bg˜(0)(x, 1) ⊂ Bg˜(t)(x, 2n+222n+1λn2 c1(n)−1)for all x ∈ M and t ∈ [0,min{T2, T̂ (n, 1, 1)}]. Scaling back and recallingthe last condition on the smallness of T˜ in (4.4), we can see that for alli ∈ {1, . . . , N} there holdsBg(0)(xi, r0) ⊂ Bg(t)(xi, 1/8).Now starting from r = r0, we apply our volume doubling estimate (b) ktimes, where k(n, λ) is the unique integer so that so that4 ≤ 2kr0 < 8.103Recalling that xi ∈ Bg(t)(y, 3/2) and the containment (4.19), the doublingestimate yields2(2n+1)kVolg(0)Bg(0)(xi, r0) ≥ Volg(0)Bg(0)(xi, 2kr0)≥ Volg(0)Bg(0)(y, 2)≥ Volg(0)Bg(t)(y, 7/4).On the other hand, the {Bg(0)(xi, r0)}Ni=1 are disjoint and each is con-tained within Bg(t)(y, 7/4) by (4.17). ThereforeN2−(2n+1)kVolg(0)Bg(t)(y, 7/4) ≤N∑i=1Volg(0)Bg(0)(xi, r0)≤ Volg(0)Bg(t)(y, 7/4)from which we can readily conclude that N ≤ 2(2n+1)k. So if we define Λ byΛ(n, λ) : = 8N ≤ 8(23n+9λn2c1(n))2n+1≤ 8(23n+9λn2 (2n+41000n)n2)2n+1,(4.21)then we can combine (4.16) and (4.18) to see that for all (y, t) ∈M × [0, T ′],there holds ∫Bg(t)(y,3/2)|Rm|n2 dVt ≤N∑i=1∫Bg(t)(xi,14)|Rm|n2 dVt≤ 4Nδv0 = 12Λδv0.(4.22)But since (M, g(t)) is a bounded curvature flow, condition (4.22) (via anapplication of Lemma 4.4.1) is in contradiction to the maximality of T ′when T ′ < min{T, T˜}. This completes the proof.4.5 Proof of Theorem DIn this section, we prove the long-time Ricci flow existence result, The-orem D. Recall the statement of this result.104Theorem. For any n ≥ 4 and v > 0, there exists c(n) > 0, ι(n, v) > 0,and ϵ(n) ≥ (10n)−10n3 such that the following holds. Suppose that (Mn, g)is any complete Riemannian manifold (not necessarily bounded curvature)with nonnegative Ricci curvature Rcg ≥ 0. Suppose that (Mn, g) has positiveasymptotic volume ratioAVR(g) ≥ v > 0,and curvature concentration satisfying∫M|Rm|n2 dVg ≤ ϵ(n)v.Then there exists a long-time solution g(t), t ∈ [0,∞) to the Ricci flow withg(0) = g which satisfies:(i) Rg(t) ≥ 0 for every t ∈ [0,∞);(ii) |Rm|g(t) ≤ 1t for every t ∈ (0,∞);(iii) injg(t) ≥ ι(n, v)√t for every t ∈ (0,∞);(iv) Volg(t)Bg(t)(x, r) ≥ c(n)v rn for every x ∈M , r > 0, t ∈ (0,∞); and(v) limt→∞ t|Rm|g(t) = 0.Before beginning the proof however, we first need to recall some of thedetails from [5]. The following is essentially taken out of the proof of theirmain theorem, with only one modification (as we will describe after).Theorem 4.5.1 ([5, Proof of Theorem 1.1]). Let (Mn, g0, x0) be some com-plete pointed noncompact Riemannian manifold of dimension n ≥ 4 (notassumed bounded curvature), which exhibits a smooth function ρ : M → Rsuch that for some C > 0 and all x ∈M , there holds:C−1dg0(x0, x)− C ≤ ρ(x) ≤ dg0(x0, x);|∇ρ(x)|2 + |∆ρ(x)| ≤ C; and∫Bg0 (x,1)|∇2ρ|n dVg0 ≤ C.(4.23)Then there exists a sequence sets of Uj ⊂M exhausting M , and metrics gjon Uj such that: each (Uj , gj) is compete with bounded curvature; for any105compact Ω ⊂ M , gj |Ω = g0|Ω for all j large enough; and for each x ∈ Uj,the metric gj satisfy the locally equivalent condition{c2x,jg0 ≤ gj ≤ C2x,jg0 on Bgj (x, 1) for someCx,j ≥ cx,j ≥ 1 satisfying(Cx,jcx,j)n ≤ 2. (4.24)Moreover, if infz∈M Rcg0(z) > −∞ then for any λ, L, v0, α, we may assumethat each of the following implications holds (by possibly deleting finitelymany of the gj):(a) If Rcg0(x) ≥ −λ for all x ∈M , then Rcgj (x) ≥ −2λ for all x ∈ Uj;(b) If Volg0Bg0(x, 2r) ≤ L · Volg0Bg0(x, r) ≤ Lv0rn for all x ∈ M andr ≤ 1, then VolgjBgj (x, 2r) ≤ 2L2 · VolgjBgj (x, r) ≤ 4L2v0rn for allx ∈ Uj, r ≤ 1/2; and(c) If∫Bg0 (x,1)|Rm|n2 dVg0 ≤ α for all x ∈M , then∫Bgj (x,1)|Rm|n2 dVgj ≤4α for all x ∈ Uj.Proof. Properties (b) and (c) are taken directly from [5, Claims 4.2, 4.4],where we have assumed the κ there is small enough so that (1 + c2κ)n ≤ 2,which is valid because c2 was an absolute constant. So it remains to seethat implication (a) holds in our setting. Note that in [5], property (a) isreplaced with the synonymous implication on scalar curvature Rg0 ≥ −λ⇒Rgj ≥ −2λ. We now describe how this modification is achieved.Using the notation from [5] of gj = gR,0 = e2FRg0, where FR = F(ρR) forR sufficiently large (depending on g0), we can compute the Ricci curvatureof gR,0 = gj in normal coordinates xi at the point p (with respect to g0) by106applying the conformal change formula:RcgR,0(∂i, ∂i) = Rcg0(∂i, ∂i)− (n− 2) (∇g0i ∇g0i FR −∇g0i FR∇g0i FR)− (∆g0FR + (n− 2)|dFR|2g0) g0(∂i, ∂i)≥ −λg0(∂i, ∂i)− C(n)[|F′′|R2∣∣∣∣ ∂ρ∂xi∣∣∣∣2 + |F′|R∣∣∣∣ ∂2ρ∂xi∂xi∣∣∣∣+ |F′|2R2∣∣∣∣ ∂ρ∂xi∣∣∣∣2]− (∆g0FR + (n− 2)|dFR|2g0) g0(∂i, ∂i)≥ −λg0(∂i, ∂i)− C(n)[ |F′′|R2|∇g0ρ|2 + |F′|R|∆g0ρ|+ |F′|2R2|∇g0ρ|2]≥ −λg0(∂i, ∂i)− C(n, g0)Re2FRg0(∂i, ∂i)=(−e−2FRλ− C(n, g0)R)e2FRg0(∂i, ∂i) ≥ −2λgR,0(∂i, ∂i).Here we have assumed R ≥ C(n, g0)λ, and have used the nonnegativity ofF, that exp(−kF)F(k) is uniformly bounded for k = 1, 2 (by an absoluteconstant), and that |∇g0ρ|, |∆g0ρ| are uniformly bounded (by a constantdepending on g0). We direct the reader to [5] for more details.We will be applying Theorem 4.5.1 with the underlying manifold (M, g0)having nonnegative Ricci curvature and positive asymptotic volume ratio. Inorder to even get started, we clearly we must first demonstrate the existenceof a distance-like function ρ with the desired properties.Lemma 4.5.2. Let (Mn, g, x0), n ≥ 4 be a complete pointed Riemannianmanifold with Rcg ≥ 0, AVR(g) > 0, and |Rm|g ∈ Ln2 (M, g). Then thereexists a smooth ρ :M → R and a constant C > 0 such that condition (4.23)holds uniformly for all x ∈M .Proof. First note by Corollary 2.5.8 and Lemma 2.2.1, there holdsν¯(M, g) ≥ log(C1(n)−1AVR(g)) =: −A > −∞.Now since |Rm|g ∈ Ln2 , [5, Lemma 4.2] implies the existence of some r ∈(0, 1] such thatsupx∈M∫Bg(x,r)|Rm|n2 dVg ≤ δ(n,A, 1)n2where δ(n,A, 1) is from [5, Theorem 2.1]. Then that result applied to thescaled up metric g˜ = r−2g implies the existence of a smooth g˜-distance-like function ρ˜ which satisfies the properties in (4.23) relative to g˜ for some107C > 0. Then by taking ρ = rρ˜ we can see that ρ is a g-distance-like functionsatisfying (4.23) relative to g for some (possibly different) C > 0, albeit withthe integral condition replaced by∫Bg(x,r)|∇2ρ|n dVg ≤ C.Now for any x ∈ M , let {xi}Ni=1 ⊂ Bg(x, 1) be some set of points such that{Bg(xi, r/2)}Ni=1 are mutually disjoint but so thatN⊔i=1Bg(xi, r/2) ⊂ Bg(x, 2) and Bg(x, 1) ⊂N⋃i=1Bg(xi, r).Then by the assumption of Rcg ≥ 0 and volume comparison, we haveωn2n ≥ VolgBg(x, 2) ≥N∑i=1VolgBg(xi, r/2) ≥ NωnAVR(g)(r/2)n.Therefore by the nature of the covering, there holds∫B(x,1)|∇2ρ|n dVg ≤N∑i=1∫Bg(x,r)|∇2ρ|n dVg≤ NC ≤ 4nr−nAVR(g)−1C.This completes the proof of the lemma.We therefore have the following approximation result with bounded cur-vature metrics, from which we will readily apply our short-time existenceresult Theorem 4.4.2.Proposition 4.5.3. For any n ≥ 4 and v > 0, there exists a dimensionalconstant C(n) ≥ 1 such that the following holds. Let (Mn, g) be a completenoncompact Riemannian manifold (not assumed bounded curvature) withRcg ≥ 0, AVR(g) ≥ v, and ∫M|Rm|n2 dVg ≤ αfor some α > 0. Then there exists a sequence sets of Uj ⊂ M exhaustingM , and metrics gj on Uj such that each (Uj , gj) is compete with boundedcurvature and for any compact Ω ⊂ M , gj |Ω = g|Ω for all j large enough.Moreover each of the following conditions holds uniformly for every x ∈ Uj:108(i) Rcgj (x) ≥ −1;(ii) VolgjBgj (x, 2r) ≤ 22n+1VolgjBgj (x, r) ≤ 22n+2ωnrn for each r ≤ 4;(iii)∫Bgj (x,4)|Rm|n2 dVgj ≤ 4α;(iv)VolgjBgj (x,r)rn ≥ ωn2 v for each r ≤ 4; and(v) Sob(Bgj (x, r), gj) ≤[C(n)(VolgjBgj (x,r)rn)− 2n, 0]for each r ≤ 4.Proof. By the scale invariance of the hypotheses and each of the conclusionssave (i), we may prove the theorem with g0 =132g which allows us to onlyrequire that conditions (ii), (iii), (iv), (v) only hold for scales r ≤ 1/2(or larger). Clearly Lemma 4.5.2 allows us to apply Theorem 4.5.1, whichgives the existence of the (Uj , gj). Moreover, properties (i), (ii), (iii) followimmediately from Theorem 4.5.1 as well because Rcg0 ≥ 0 and volumecomparison impliesVolg0Bg0(x, 2r)(2r)n≤ Volg0Bg0(x, r)rn≤ ωnfor all r > 0. Now fix some j ∈ N, x ∈ Uj , r ∈ (0, 1] and write VolgjBgj (x,r)rn =vxjr. By the locally equivalent condition (4.24) and the fact that AVR(g0) ≥v, we obtainvxjr ≥ cnx,jVolg0Bgj (x, r)rn≥ cnx,jVolg0Bg0(x,C−1x,j r)rn≥ 12Volg0Bg0(x,C−1x,j r)(C−1x,j r)n≥ ωn2v,which proves (iv). Now it remains to show (v). Just as above, we mayobtain an upper volume ratio bound:vxjr ≤ Cnx,jVolg0Bgj (x, r)rn≤ Cnx,jVolg0Bg0(x, c−1x,jr)rn≤ 2Volg0Bg0(x, c−1x,jr)(c−1x,jr)n.Because Rcg0 ≥ 0 and C−1x,j r ≤ c−1x,jr, we may once again use volume com-parison to say that12Volg0Bg0(x, c−1x,jr)(c−1x,jr)n≤ vxjr ≤ 2Volg0Bg0(x, c−1x,jr)(c−1x,jr)n109which is equivalent toVolg0Bg0(x, c−1x,jr)(c−1x,jr)n∈[12vxjr, 2vxjr].Now by Lemma 2.5.2, we obtain the Sobolev inequalitySob(Bg0(x, rc−1x,j), g0) ≤[C1(n)(vxjr2)− 2n, 0]for some dimensional constant C1(n) ≥ 1. Since Bgj (x, r) ⊂ Bg0(x, rc−1x,j),we may again apply the locally equivalent condition (4.24) to obtain Sobolevinequality for gj :(∫Bgj (x,r)|u| 2nn−2 dVgj)n−2n≤(Cx,jcx,j)n−2C1(n)(vxjr2)− 2n∫Bgj (x,r)|∇u|2 dVgj≤ 2C1(n)v−2nxjr∫Bgj (x,r)|∇u|2 dVgjfor all u ∈W 1,20 (Bgj (x, r)). This gives (v) and completes the proof.We now prove Theorem D by a diagonal blow-down argument.Proof of Theorem D. Given n ≥ 4 and v ∈ (0, 1], let (Mn, g) satisfy thehypotheses with this choice of n, v and fix some x0 ∈ M . For any R ≥ 1,denote gR = R−2g, which still satisfies AVR(gR) ≥ v and∫M|Rm|n2 dVgR ≤ ϵ(n)v.Then by Proposition 4.5.3, we obtain a sequence of sets UR,j ⊂M exhaustingM and complete bounded curvature metrics gR,j on UR,j which satisfy gR,j =gR on BgR(x0, 1). Moreover each of the following conditions holds uniformlyfor every x ∈ UR,j :(i) RcgR,j (x) ≥ −1;(ii) VolgR,jBgR,j (x, 2r) ≤ 22n+1VolgR,jBgR,j (x, r) ≤ 22n+2ωnrn for each r ≤4;110(iii)∫BgR,j (x,4)|Rm|n2 dVgR,j ≤ 4ϵ(n)v;(iv)VolgR,jBgR,j (x,r)rn ≥ ωn2 v for each r ∈ (0, 1]; and(v) Sob(Bgj (x, r), gj) ≤[C1(n)(VolgjBgj (x,r)rn)− 2n, 0]for each r ∈ (0, 4].Writing v0 =ωn2 v, we replace condition (iv) withvxRjr :=VolgR,jBgR,j (x, r)rn≥ v0 for each r ∈ (0, 1], (iv’)and replace condition (v) withSob(Bgj (x, 4r), gj) ≤[42C1(n)v− 2nxRjr, 0]for each r ∈ (0, 1]. (v’)If we write λ = 42C1(n), then we can extract from Theorem 4.4.2 the exis-tence of some constants C2, c1, ι,Λ, T˜ based on this choice of λ and v0. Nowif we takeϵ(n) =δ(n, λ)ωn8, (4.25)then we may apply Theorem 4.4.2 to each metric gR,j . The result is afamily of Ricci flows {gR,j(t)}, each defined on [0, T˜ ] (note that this timeinterval depends only on n, λ, v0 and not on j, R). Taking the diagonalscale-normalized sequence gj = j2gj,j , the result is a sequence of flows gj(t),t ∈ [0, j2T˜ ] such that gj(0) = g on Bg(x0, j) and for each j ∈ N, all of thefollowing are satisfied for every (x, t) ∈M × [0, j2T˜ ]:(i) Rgj(t)(x) ≥ −nj−2;(ii) |Rm|gj(t)(x) ≤ 1t ;(iii) injgj(t)(x) ≥ ι(n, v)√t;(iv) Volgj(t)Bgj(t)(x, r) ≥ c1(n)v rn for all r ∈ (0, j];(v)∫Bgj(t)(x,j)|Rm|n2 dVgj(t) ≤ Λ(n, λ)δ(n, λ)v0; and(vi) Sob(Bgj(t)(x, j), gj(t)) ≤ [C2(n)λv− 2n0 ,1j2].111Then just as in the proof of [5, Theorem 1.1], we may apply [11, Corollary3.2], Shi’s modified interior estimates [16, Theorem 14.16], the Arzela`-AscoliTheorem, and a diagonal argument, to conclude that some subsequenceof gj(t) converges in C∞loc(M × [0,∞)) to a long-time Ricci flow g(t) withg(0) = g satisfying the following for all t ∈ [0,∞):(i) Rg(t) ≥ 0;(ii) |Rm|g(t) ≤ 1t ;(iii) injg(t) ≥ ι(n, v)√t;(iv) Volg(t)Bg(t)(x0, r) ≥ c1(n)v rn for all r ∈ (0,∞);(v)∫M |Rm|n2 dVg(t) ≤ Λ(n, λ)δ(n, λ)v0; and(vi) Sob(M, g(t)) ≤ [C2(n)λv−2n0 , 0].Finally, obtaining the faster than 1/t curvature decay follows by applyingTheorem A to the bounded curvature metric g(1), which is validated byour smallness assumption on δ(n, λ) in (4.12). This completes the proof ofTheorem D.Remark 4.5.4. By recalling our definitions of ϵ(n) in (4.25), of δ(n, λ) in(4.5), and of Λ(n, λ) in (4.21), as well as the fact that λ ≤ 32(10n)3n byRemark 2.5.7, we obtain the explicit boundϵ(n) ≥ ωn8 18(23n+9[32(10n)3n]n2 (2n+41000n)n2)2n+1( 1107n3[32(10n)3n])n2≥ (10n)−10n3 .112Chapter 5Removing Scalar CurvatureDependenceIn this chapter, we will prove the results given in Section 1.4, which areprimarily concerned with generalizing the main results from [5] by removingthe scalar curvature bound assumption from their hypotheses. The layoutof this chapter is as follows.• In Section 5.1, we give modified distance and volume distortion esti-mates that remove the scalar curvature assumption.• In Section 5.2, we prove a localized Ricci flow smoothing result (Theo-rem E) that is very similar - both its statement and proof - to Theorem4.4.2, albeit without a lower scalar or Ricci curvature bound assump-tion.• In Section 5.3, we prove Theorem F which is a Ricci flow smoothingresult for metric measure spaces that are not necessarily Riemannianmanifolds.• In Section 5.4, we prove Theorem G which gives topological and vol-ume rigidity for manifolds which have very small global curvature con-centration. This result is similar in flavour to the topological rigiditygiven in Theorem A and our gap theorem for nonnegative Ricci cur-vature manifolds, Theorem C.5.1 Modified distance distortionIn this section, we will give a modified version of the Expanding BallsLemma which removes the dependence on a lower scalar curvature bound.This will be one of the key steps towards proving Theorem E, which in turnis the key result in the proofs of Theorems F and G.113First, we recall the well-known Shrinking Balls Lemma due to Simon-Topping, which dictates how quickly distances can contract under Ricci flowswith scale-invariant curvature bounds:supy∈Bg(0)(x,r)|Rm|g(t)(y) ≤ c0/t =⇒ Bg(t)(x, r − β√c0t) ⊂ Bg(0)(x, r)Here β = β(n) is a dimensional constant, which is now fixed notation inthe remainder of the chapter. We direct the reader to [41, Corollary 3.3], orLemma 4.2.1 for a precise statement.Obtaining control on how quickly distances can expand under the flowusually requires control on the negative part of the Ricci or scalar curvaturealong the flow. See [41, Lemma 3.1], [24, Lemma 3.7], [29, Lemma 2.2], orLemma 4.2.2 for example. In our setting, however, we are able to substitutethis L∞ curvature assumption for control on the local curvature concentra-tion along the flow. We begin with an elementary lemma that allows us tocontrol the rate at which volumes are perturbed, which will be useful in itsown right.Lemma 5.1.1 (Volume Perturbation). For any n ∈ N, v, σ, r > 0, and anypointed (not assumed complete) Ricci flow (Mn, g(t), x0), t ∈ [a, b] for somea < b, let Ω ⊂⊂M be any precompact set satisfying:(a) Bg(t)(x0, r) ⊂ Ω for each t ∈ [a, b];(b)(∫Ω |R|n2 dVg(t)) 2n ≤ σ for all t ∈ [a, b];(c) Volg(t)Bg(t)(x0, r) ≥ vrn for all t ∈ [a, b].Then whenever a ≤ s ≤ t ≤ b, there holdse−σv− 2n r−2(t−s)Volg(s)Ω ≤ Volg(t)Ω ≤ eσv− 2n r−2(t−s)Volg(s)Ω.Remark 5.1.2. Our most common application of Lemma 5.1.1 will be witha = s = 0 and Ω = Bg(0)(x0, 2r). In this setting, if we further assume thatthe flow enjoys an α/t curvature bound, then we can see (by the ShrinkingBalls Lemma) that condition (a) can be replaced by the assumption thatBg(0)(x0, 2r) ⊂ Ω as long as we also restrict b ≤ r2β2α .114Proof of Lemma 5.1.1. By the evolution of the volume form under the Ricciflow ∂tdVg(t) = −Rg(t)dVg(t), the assumed smallness of integral scalar cur-vature (b) and the lower bound of volumes at time t in (c), we obtain∂tVolg(t)(Ω) ≤(∫Ω|R|n2 dVg(t)) 2n[Volg(t)(Ω)]n−2n≤ σ[Volg(t)(Ω)]−2nVolg(t)(Ω)≤ σ[Volg(t)Bg(t)(x0, r)]−2nVolg(t)(Ω)≤ σ(vrn)− 2nVolg(t)(Ω).Therefore by comparison, we obtain Volg(t)Ω ≤ eσv− 2n r−2(t−s)Volg(s)Ω for any0 ≤ s ≤ t ≤ T . The other inequality follows from an identical argument:∂tVolg(t)(Ω) ≥ −(∫Ω|R|n2 dVg(t)) 2n[Volg(t)(Ω)]n−2n≥ −σVolg(t)(Ω)[Volg(t)(Ω)]−2n≥ −σVolg(t)(Ω)[Volg(t)Bg(t)(x0, 1)]−2n≥ −σ(vrn)− 2nVolg(t)(Ω).This volume perturbation estimate allows us to remove the assumptionof a lower bound on scalar curvature in the version of the Expanding BallsLemma from Section 4.2 (Lemma 4.2.2). The following is the main result ofthis section.Lemma 5.1.3 (Expanding Balls). For any n ∈ N and any α, v, σ > 0, thereexists T̂ (n, α, v, σ) > 0 such that the following holds. Let (Mn, g(t), x0),t ∈ [0, T ] be a pointed Ricci flow (not assumed complete) such that for someV > 0 there holds:(a) Bg(t)(x0, 2n+2V v−1) ⊂⊂M for all t ∈ [0, T ];(b)(∫Bg(0)(x0,2)|R|n2 dVg(t)) 2n ≤ σ for all t ∈ [0, T ];(c) supx∈Bg(0)(x0,2)|Rm|g(t)(x) ≤ αt−1 for all t ∈ (0, T ];115(d) Volg(0)Bg(0)(x0, 2) ≤ V ; and(e) Volg(t)Bg(t)(x, r) ≥ vrn whenever x ∈ Bg(0)(x0, 1), r ∈ (0, 12 ], andt ∈ (0, T ].Then for all t ∈ [0,min{T, T̂}], we have Bg(0)(x0, 1) ⊂ Bg(t)(x0, 2n+2V v−1).Proof. The proof is nearly identical to the argument in Section 4.2. Theonly changes are as follows. First, the maximal time interval in our settingwill beT̂ = min{v2n log 2σ,14β2α}.Now the only setting in which the lower bound of scalar curvature was usedin Theorem 4.2.2 was to obtain the volume controlVolg(t)(Bg(0)(x0, 2)) ≤ 2Volg(0)(Bg(0)(x0, 2))for all t ∈ [0,min{T, T̂}]. For us, this implication follows immediately fromLemma 5.1.1 by our restrictions on the size of T̂ and Remark 5.1.2.As discussed in Section 1.4, if we had a version of the Expanding BallsLemma that gave a continuous rate of expansion at time zero (rather thanthe discontinuous conclusion that an initial unit ball is contained in somelarge ball for even arbitrarily small times t > 0), then we would be able toanswer in the affirmative both of Questions 1.4.3 and 1.4.4. The following isa precise formulation of such a theorem that would suffice for our purposes,with hypotheses that I suspect are sufficient for the desired conclusion.Conjecture 5.1.4 (Expanding Balls with Continuous Rate of Expansion).For any n ∈ N, v > 1, and α,A, σ, ϵ > 0, there exists T̂ (n, α,A, σ, ϵ, v) > 0such that the following holds. Let (Mn, g(t), x0), t ∈ [0, T ] be a pointedRicci flow (not assumed complete) such that:(a) Bg(t)(x0, 2) ⊂⊂M for all t ∈ [0, T ];(b)(∫Bg(0)(x0,2)|Rm|n2 dVg(t)) 2n ≤ σ for all t ∈ [0, T ];(c) supx∈Bg(0)(x0,2)|Rm|g(t)(x) ≤ αt−1 for all t ∈ (0, T ];(d) Volg(0)Bg(0)(x, r) ≤ vωnrn for all x ∈ Bg(0)(x0, 1) and r ≤ 1; and116(e) Sob(Bg(t)(x, r)) ≤ [A, 1] whenever x ∈ Bg(0)(x0, 1), r ∈ (0, 1], andt ∈ (0, T ].Then for all t ∈ [0,min{T, T̂}], we have Bg(0)(x0, 1) ⊂ Bg(t)(x0, 1 + ϵ).Here I will outline one approach that I have been considering in orderto prove the above conjecture, as well as emphasizing the current shortfallsof the argument.First, given any such constants n, α, v, A, σ, the non-continuous Expand-ing Balls Lemma 5.1.3 guarantees the existence of constants T˜ and R (de-pending only on the above constants) such thatBg(0)(x, r) ⊂ Bg(t)(x, rR)whenever x ∈ Bg(0)(x0, 1) and t ∈ [0, T˜ r2]. Then for a given ϵ > 0, setδ =ϵ4R.Moreover, for the sake of simplicity here (this could be achieved in our settingat a small scale, or by decreasing σ), we will assume that injg0(x0) > 1so the exponential map is a diffeomorphism on some open set containingBg(0)(x0, 1), and we will writeϕ : Bg(0)(x0, 1)→ BgE (0, 1)to be this diffeomorphism. In particular, ϕ takes the g(0)-geodesics thatconnect x0 to points in ∂Bg(0)(x0, 1) to straight lines to the unit spherein Rn. We will also write ◦g to be the standard Euclidean metric on theEuclidean unit sphere.Consider T ′ to be the first time in which the conclusion of the conjecturefails, so that it suffices to show that T ′ must actually be uniformly boundedbelow by some time T̂ which depends only on the given constants, and nototherwise on the flow. So we may assume there is some y ∈ ∂Bg(0)(x0, 1)such thatdg(T ′)(x0, y) = 1 + ϵand thatBg(0)(x0, 1) ⊂ Bg(t)(x0, 1 + ϵ)117for all t ∈ [0, T ′]. Consider the set of points on the g(0)-sphere ∂Bg(0)(x0, 1)that are at least δ/2, and at most δ, g(0)-distance away from x0. That is,we are considering the set of pointsSδ =[Bg(0)(y, δ) \Bg(0)(x0, δ/2)] ∩ ∂Bg(0)(x0, 1).We’re interested in showing that the average g(t)-distance from x0 to pointsx ∈ Sδ stays less than 1 + ϵ/2 for all times sufficiently small, i.e., thatA(t) :=1Area◦g(ϕ(Sδ))∫ϕ(Sδ)distg(t)(x0, x) dA◦g(x) ≤ 1 + ϵ/2 (5.1)for all t ∈ [0, T̂ ]. Note here that A(0) = 1 follows trivially from the defini-tions. For each x ∈ Sδ, let γx denote the unit speed g(0)-geodesic connectingx and x0 (we’re filling out a cone worth of geodesics from the center pointx0 to a cap on the g(0)-unit sphere). First, note that for any x ∈ Sδ, we mayapply the (non-continuous) Expanding Balls Lemma 5.1.3 at a very smallscale to conclude that for all t ∈ [0, δ2T˜ ], there holdsBg(0)(x0, δ) ⊂ Bg(t)(x0, δR) = Bg(t)(x0, ϵ/4).Thus, if we “cut off” the first g(0)-portion of the geodesics γx, in other wordswe work instead with the curvesγ˜x := γ|[δ,1],then we can say that, for any x ∈ Sδ, we havedistg(t)(x0, x) ≤ϵ4+ Lengthg(t)(γ˜x).So instead of proving the more nebulous inequality of (5.1), we can workdirectly with this family of curves to show that1Area◦g(ϕ(Sδ))∫ϕ(Sδ)Lengthg(t)(γ˜ϕ−1(x)) dA◦g(x) ≤ 1 +3ϵ4, (5.2)for all t ∈ [0, T̂ ]. For the sake of simplicity, We’ll abuse notation a bit hereby henceforth identifying x and ϕ(x). Also, we denote Cδ to be the coneformed by the collection of all γ˜x, x ∈ Sδ. Colloquially speaking, Cδ is a118very thin and hollow ice cream cone with its tip bitten off. Notice that∫ϕ(Sδ)Lengthg(t)(γ˜x) dA◦g =∫ϕ(Sδ)∫ 1δ∥γ′x(s)∥g(t) ds dA◦g=∫ϕ(Sδ)∫ 1δs−(n−1)sn−1∥γ′x(s)∥g(t) ds dA◦g=∫ϕ(Cδ)s−(n−1)∥γ′x(s)∥g(t) dx.When we differentiate in time, the derivative only lands ∥γ′x(s)∥g(t), whichproduces a curvature term, and so we can calculateddt∫ϕ(Sδ)Lengthg(t)(γx) dA◦g ≤ C(n)∫ϕ(Cδ)s−(n−1)|Rm|(x, t) dx≤ C(n, δ)∫ϕ(Cδ)δ−(n−1)|Rm|(x, t) dx.Since the above integral is just the integral in the underlying coordinatesystem coming from ϕ, we can write the volume form for the metric g(t) inthis coordinate system asdVg(t) =√detϕ(g(t))ij dx =: f(t)dx .Thus we can write∫ϕ(Cδ)|Rm|g(t) dx=∫ϕ(Cδ)f(t)−2n f(t)2n |Rm|g(t) dx≤(∫Bg(0)(x0,1)|Rm|n2 (x, t) dVg(t)) 2n(∫ϕ(Cδ)f(t)−2nnn−2 dx)n−2n.The first term in the latter product is bounded by σ by assumption. Wewould like to be able to say that the second term likewise can be boundedas well, depending only on the given constants. In other words, we wouldlike the following to be true:(∫ϕ(Cδ)f(t)−2nnn−2 dx)n−2n≤ C(n, α, v, A, σ, ϵ) (5.3)119If it were known that (5.3) is true for all t ∈ [0, T̂ ], we could conclude that(5.2), and thus (5.1) as well, by possibly shrinking T̂ further depending onC and δ.Now, for the sake of contradiction, assume thatT ′ ≤ min{T̂ ,δ24T˜}.Since, at time T ′ in particular, (5.1) holds, we may find some xT ′ such that1 +ϵ2≥ Lengthg(T ′)(γxT ′ ) ≥ dg(t)(x0, xT ′).But we also know thatδ2≤ dg(0)(y, xT ′) ≤ δ,which, in particular, implies (by the non-continuous Expanding Balls Lemma5.1.3) that for all t ∈ [0, δ2T˜ /4], we havedg(T ′)(y, xT ′) ≤ δR =ϵ4.But then, the triangle inequality would imply the following absurdity:1 + ϵ = dg(T ′)(x0, y) ≤ dg(T ′)(x0, xT ′) + dg(T ′)(y, xT ′) ≤ 1 +ϵ2+ϵ4.To wrap up this section, we reiterate here that in order to prove Con-jecture 5.1.4, which would in turn provide affirmative answers to Questions1.4.3 and 1.4.4, it would suffice to prove inequality (5.3). There are slightalterations to the above outline as well, where for instance, one may workwith the evolving normal coordinate (this may make proving (5.3) easier, atthe expense of added difficulty elsewhere).5.2 Localized Ricci flow smoothingIn this section we present the proof of Theorem E, which is a short-timeRicci flow existence and smoothing result for manifolds that have locally(uniformly) bounded Sobolev data and locally (uniformly) small curvatureconcentration. First, we recall the precise statement of Theorem E.120Theorem. For all n ≥ 4, A, τ, v0 > 0, there are C0, σ, T > 0 dependingonly on n,A, τ, v0 such that the following holds. Let (Mn, g) be a completenoncompact manifold (not necessarily bounded curvature) such that for someϵ ≤ σ and all x ∈M , the metric g satisfies:(i) VolgBg(x, r) ≤ v0rn for all r ≤ 1;(ii) ν¯(Bg(x, 1), g, τ) ≥ −A;(iii)(∫Bg(x,1)|Rm|n2 dVg) 2n ≤ ϵ; and(iv) infx∈MRcg(x) > −∞.Then there exists a complete Ricci flow g(t) on M × [0, T ] with g(0) = gsuch that for all (x, t) ∈M × (0, T ], there holds|Rm|g(t)(x) ≤ C0ϵt−1,injg(t)(x) ≥ C−10√t, and(∫Bg(0)(x,116)|Rm|n2 dVg(t)) 2n≤ C0ϵ.(5.4)We claim that it suffices to show that the theorem holds for completebounded curvature metrics, albeit for slightly different scales (this will beClaim 5.2.1, see below). The reason why this is sufficient is as follows.We know by [5, Theorem 2.1] that there exists a smooth proper functionρ :M → R which, for some constant C depending on the metric g, satisfiesthe following conditions uniformly for all x ∈M :C−1dg(x0, x)− C ≤ ρ(x) ≤ dg(x0, x);|∇ρ(x)|2g + |∆gρ(x)| ≤ C; and∫Bg(x,1)|∇2ρ|n dVg ≤ C.(5.5)We emphasize again here that the ineffective lower Ricci curvature bound(iv) is only used to show that (5.5) holds. Once we know that (5.5) holds,by the proof of [5, Theorem 1.1] (summarized in Theorem 4.5.1), we can saythat there exists a sequence of exhausting precompact open sets Uj ⊂M andcomplete bounded curvature metrics gj on Uj that satisfy all the following:• For any compact Ω ⊂M , gj = g on Ω for all j sufficiently large;• VolgjBgj (x, r) ≤ 2v0rn for all r ≤ 1/2;121• ν¯(Bgj (x, 1/2), gj , τ) ≥ −A˜(n,A, τ); and•(∫Bgj (x,1)|Rm|n2 dVgj) 2n ≤ 4ϵ.Define the sequence of blown-up metrics g˜j = 162gj , which (in particular)satisfies each of the following conditions uniformly with respect to j ∈ N andx ∈ Uj :(I) Volg˜jBg˜j (x, r) ≤ 2v0rn for all r ≤ 8;(II) ν¯(Bg˜j (x, 4), g˜j , τ) ≥ −A˜; and(III)(∫Bg˜j (x,4)|Rm|n2 dVg˜j) 2n≤ 4ϵ.Now we will show that the g˜j actually satisfy the conclusions of the followingclaim, which is only a slightly modified version of Theorem 4.4.2.Claim 5.2.1. There exists T˜ , C0, σ depending only on n,A, τ, v0 such thatthe complete bounded curvature Ricci flow g˜j(t) with g˜j(0) = g˜j exists atleast until time T˜ and satisfies|Rm|g˜j(t)(x) ≤ C0ϵt−1,injg˜j(t)(x) ≥ C−10√t, and(∫Bg˜j(0)(x,54) |Rm|n2 dVg˜j(t)) 2n≤ C0ϵuniformly for all (x, t) ∈ Uj × [0, T˜ ], assuming that ϵ ≤ σ.To prove the theorem (assuming the claim is true), we briefly sketch theargument of the proof of [5, Theorem 1.1] (also similar to the end of theproof of Theorem D). Apply the claim to each g˜j and scale back down toobtain a sequence of flows gj(t) on Uj× [0, T ], where T = T˜162 , and each gj(t)satisfies the estimates (5.4). Since gj = g on any compact set as j → ∞,we can say by [11, Corollary 3.2] and Shi’s modified interior estimates [16,Theorem 14.16] thatlim supj→∞supΩ×[0,T ]|∇kRm|gj(t) ≤ C(n, k,Ω, g),for any k ≥ 0 and any compact Ω ⊂ M . Then we may apply the Arzela´-Ascoli Theorem and a diagonal argument on a countable covering of coor-dinate charts to conclude that some subsequence of the gj(t) converges in122C∞loc(M × [0, T ]) to a Ricci flow g(t) on M × [0, T ] with g(0) = g. More-over, g(t) satisfies (5.4) (that is, the desired properties of our flow as in thetheorem) because each gj(t) does, and is thus also a complete flow by theShrinking Balls Lemma. Therefore it suffices to prove the claim.Proof of claim. This argument is only a mild modification the proof of The-orem 4.4.2 presented in Section 4.4. We will follow the steps of that proofhere, using the same notation. For brevity, we will suppress the dependenceon j by denoting g(t) = g˜j(t), t ∈ [0, T ] to be the unique bounded curva-ture Ricci flow emanating from g˜j . Let us begin by recalling the outlineof the proof from Section 4.4. There we defined some constants T˜ ,Λ de-pending only on n,A, τ, v0 (there, the values of v0 and τ were fixed as 2ωnand 2 respectively), and then defined T ′ > 0 to be the maximal time in[0,min{T, T˜}] for which∫Bg(t)(x,1)|Rm|n2 dVg(t) ≤ Λϵn2 (5.6)uniformly for all (x, t) ∈M × [0, T ′]. Then the proof is split into three stepswhich we now briefly describe.(Step 1) For any α > 0, let T1 denote the largest time in which |Rm|g(t) ≤αt , which is necessarily positive because the flow g(t) is of boundedcurvature. Then this step shows that the flow enjoys a local L2-Sobolevinequality, and thus also a uniform lower bound on volume ratios up toa fixed scale, for all (x, t) ∈ M × [0,min{T ′, T1}] where the constantsare uniform with respect to the choice of (x, t).(Step 2) This step shows that the T1 from Step 1 can actually be taken to onlydepend on n,A, τ, v0 as long as Λ is sufficiently large (depending onlyon n,A, τ, v0). Thus we assume that T˜ ≤ T1. Note that in the proofin Section 4.4, the choice of α was fixed, while here it should dependon ϵ.(Step 3) By Steps 1 and 2, we see that the desired conclusions of the theoremhold on the interval [0, T ′]. Thus in this step it is shown that in factT ′ can be taken to be uniformly bounded from below, only dependingon n,A, τ, v0. This is the step in which the Expanding Balls Lemma isused to say that there must exist some r0 > 0 such that Bg(0)(x, r0) ⊂Bg(t)(x, 1) for all (x, t) ∈ M × [0, T˜ ]. This containment allows oneto leverage the evolution of the curvature concentration against the123known volume bounds at time t = 0 to derive a contradiction if Λ islarge enough and T ′ < T˜ .We now describe the necessary changes to the proofs of each of thesethree steps in our setting here. For clarity of the proof, instead of definingthe constants T˜ ,Λ, σ ahead of time, we will simply start with them all beingequal to 1 and adjust them as we go only ever depending on n,A, τ, v0.Our first significant change is in the definition of T ′. Rather than using thedefinition in (5.6), we define T ′ to be the maximal time in [0,min{T, T˜}]such that ∫Bg(0)(x,54)|Rm|n2 dVg(t) ≤ Λϵn2 (5.7)uniformly for all (x, t) ∈M× [0, T ′]. This subtle change of the domain beingdefined with respect to g(0) rather than g(t) allows for the application ofour version of the Expanding Balls Lemma (Lemma 5.1.3). Despite thischange, we can still say that T ′ > 0 because g(t) is a bounded curvatureflow (this argument follows by a slight modification of Lemma 4.4.1). Thusby restricting T˜ and σ sufficiently small, Step 1 of the proof in Section4.4 then follows synonymously. In particular, we obtain a uniform localL2-Sobolev inequality along the flow:Sob(Bg(t)(x, 1), g(t)) ≤ [CS(n,A, τ), 1] (5.8)for all (x, t) ∈M×[0,min{T ′, T1}]. Also recall that an L2-Sobolev inequalityimplies a lower bound on volume ratios (see Lemma 2.5.1):{Volg(t)Bg(t)(x, r) ≥ v˜(n,CS)rn for all(x, t) ∈M × [0,min{T ′, T1}], r ∈ (0, 1].(5.9)Recall that in (5.8) and (5.9), T1 is the largest time in which the flowg(t) enjoys an α/t curvature bound, with α = C0ϵ (here C0 will be definedlater in terms of Λ). Since g(t) is a bounded curvature flow, we know thatT1 > 0, though a priori it appears to depend on the bound on the curvatureand not merely the constants we are working with. In Step 2, however, weshow that T1 can be assumed to be at least as large as T′ whenever T˜ andσ are chosen small enough. It is necessary in this step to diverge from thelimiting contradiction argument presented in Section 4.4. Instead, we utilizea more classical argument, in which we argue by contradiction by assumingthat T1 < T′. By the maximality of T1, the flow certainly does enjoy anα/t curvature bound on [0, T1], and thus condition (5.7) implies (5.6) on this124interval by the Shrinking Balls Lemma, assuming T˜ is small enough. At thispoint, we may directly apply [4, Proposition 2.1] to derive a contradiction ifσ is chosen small enough. Note that this argument, and constants thereof,are independent of the exact choice of ϵ ≤ σ.It remains to show that Step 3 goes through as well. By taking T˜ and σsmall enough, we may still arrive at the conclusion that∫Bg(t)(x,1/4)|Rm|n2 dVg(t) ≤ 4 · (4ϵ)n2holds uniformly for all (x, t) ∈M × [0, T ′]. Now by our modified assumptionon the smallness of the local curvature concentration in (5.7), our assumedinitial volume upper bound (I), and our lower bound on volume ratios alongthe flow in (5.9), we can say by our version of the Expanding Balls Lemma(Lemma 5.1.3) that if T˜ is small enough, then there is some r0(n,A, τ, v0) > 0such that for all (x, t) ∈M × [0, T ′], there holds∫Bg(0)(x,r0)|Rm|n2 dVg(t) ≤ 4 · (4ϵ)n2 .Now the remainder of Step 3 follows verbatim to the proof in Section 4.4because by (I) and (5.9) we have a uniform doubling estimate on scales lessthan 4:Volg(0)Bg(0)(x, 2r) ≤ 23n+1v0v˜−1Volg(0)Bg(0)(x, r)≤ (23n+1v0v˜−1)v0rn for all r ∈ (0, 4].Thus we obtain a contradiction if Λ is large enough and if T ′ is assumed tobe strictly less than min{T, T˜}. The constant C0 can therefore be taken tobe Λ2n . For convenience in the proof of Theorem F, we will further assumethat σ is small enough so that C0σ <14(n−1) .5.3 Smoothing of measuresIn this section, we present the proof of Theorem F, which is a smoothingresult for metric measure space limits of smooth manifolds that satisfy thehypotheses of Theorem E for some choice of constants A, τ, v0. Alternatively,this result can be viewed as a Gromov-Hausdorff compactness theorem for125the space of such smooth manifolds, as in the statement of Theorem E. Let’srecall the precise (and lengthy) statement.Theorem. For any n ≥ 4, A, τ, v0 > 0, let σ,C0, T be as in Theorem E,which all depend only on n,A, τ, v0. Let (Mni , gi, pi) be any sequence ofcomplete pointed Riemannian manifolds (not necessarily bounded curvature)each satisfying the hypotheses(i) VolgBg(x, r) ≤ v0rn for all r ≤ 1;(ii) ν¯(Bg(x, 1), g0, τ) ≥ −A;(iii)(∫Bg(x,1)|Rm|n2 dVg) 2n ≤ σ; and(iv) infx∈MRcg(x) > −∞.Then there exists a smooth pointed manifold (M∞, x∞) endowed with a com-plete distance metric d0 (which generates the same topology as the smoothstructure of M∞) and a Radon measure µ such that, up to a subsequence,(Mi, dgi ,Volgi , pi) −→ (M∞, d0, µ, x∞) (5.10)where the convergence is as pointed metric measure spaces (as describedprecisely below). Moreover, there exists a complete Ricci flow g(t) on M∞×(0, T ] satisfying the estimates|Rm|g(t)(x) ≤ C0σt−1,injg(t)(x) ≥ C−10√t, and(∫Bg(0)(x,116)|Rm|n2 dVg(t)) 2n≤ C0σ,which emerges from (M∞, µ) in the weak sense, i.e., that for any precompactBorel set E ⊂M∞, there holdslimt→0Volg(t)(E) = µ(E).The notion in which the convergence (5.10) holds is as follows. There is anexhausting sequence Ωk ⊂ M∞ of open sets, and functions Fk : Ωk → Mik(each of which are diffeomorphisms onto their images) with Fk(x∞) = pikwhich satisfylimk→∞VolF ∗k gik (E) = µ(E)126for any precompact Borel set E ⊂M∞, and(Mik , dgik , pik)pointed Gromov-Hausdorff−−−−−−−−−−−−−−−−→ (M∞, d0, x∞),where the Fk’s may serve as the approximating functions.For each i ∈ N, let gi(t), t ∈ [0, T ] be the Ricci flow emanating from gigiven by Theorem E (note that T here is independent of i). Without lossof generality, we will assume that T ≤ 1, and for convenience of notationwe denote α = C0σ which satisfies α <14(n−1) by the proof of Theorem E.Thus the gi(t) satisfy all the following conditions, uniformly for all (x, t) ∈Mi × [0, T ]:Volgi(0)Bgi(0)(x, r) ≤ v0rn for all r ≤ 1;Volgi(t)Bgi(t)(x, r) ≥ v˜(n,A, τ)rn for all r ≤ 116 ;|Rm|gi(t)(x) ≤ αt ;injgi(t)(x) ≥ C−10√t; and(∫Bgi(0)(x,116) |Rm|n2 dVgi(t)) 2n ≤ α.(5.11)By Hamilton’s Compactness Theorem [23] (and a diagonal argument),some subsequence of the (Mi, gi(t)), t ∈ (0, T ] converges in the Cheeger-Gromov sense to a smooth pointed complete Ricci flow (M∞, g(t), x∞), t ∈(0, T ], that also satisfies the properties in (5.4). For the sake of brevity, wewill continue to write (Mi, gi, pi) for this subsequence, i.e., relabel so thatik = i. The functions Fi in the statement of the theorem are precisely thosecoming from the definition of the Cheeger-Gromov convergence. That is tosay that Fi : Ωi →Mi, which are diffeomorphisms onto their images, whereΩi ⊂ Ωi+1 ⊂ M∞ is an exhausting sequence of open precompact sets withF ∗i gi(t)→ g(t) in C∞loc(M∞×(0, T ]). Thus it suffices to prove the existence ofthe distance metric d0 and measure µ as well as their convergence properties.We will deal with d0 first.5.3.1 Existence and properties of d0To show that this family of Ricci flows gi(t) are regular enough to extracta Gromov-Hausdorff limiting distance metric at time zero, we will use thefollowing claim, which says that the conclusion of [26, Theorem 2.4] holds inthis setting, i.e., that the flows gi(t) are uniformly (with respect to i) locallybi-Ho¨lder.127Claim 5.3.1. There exists some C ≥ 1 depending only on n,A, τ, v0 suchthat for any xi, yi ∈Mi with dgi(xi, yi) ≤ 1 and t ∈ [0, T ], there holdsC−1d11−2(n−1)αgi(t)(xi, yi) ≤ dgi(xi, yi) ≤ Cd11+2(n−1)αgi(t)(xi, yi). (5.12)By an identical argument to [4, Theorem 1.4] and [5, Theorem 5.1], theclaim implies the existence of a distance metric d0 on M∞ such that somefurther subsequence of the gi (still calling them gi) satisfieslimi→∞dF ∗i gi(x, y) = d0(x, y) (5.13)for every pair of points x, y ∈ M∞. Thus we can say that the bi-Ho¨lderproperty (5.12) descends to the limit as well:C−1d11−2(n−1)αg(t) (x, y) ≤ d0(x, y) ≤ Cd11+2(n−1)αg(t) (x, y) (5.14)whenever d0(x, y) ≤ 1. This implies that d0 generates the same topology asM∞. The fact that (M∞, d0) is a complete metric space follows from (5.14)by an argument that we now sketch. If (M∞, d0) were not complete thenwe could choose some continuous rectifiable curve γ : [0, 1) → M∞, i.e.,Lengthd0(γ) <∞, but with the property thatlims→1−γ(s) ̸∈M∞.Now we may pick ϵ > 0 small enough so that dg(t)(x, y) ≤ ϵ implies thatd0(x, y) ≤ 1 by the right-hand side of (5.14). Since g(T ) is complete,we may choose countably many points {xj}j∈N ⊂ γ([0, 1)) such that the{Bg(T )(xj , ϵ)}j∈N are mutually disjoint, and we may let {tj}j∈N ⊂ (0, 1) becorresponding times with the property that tj is the latest time in whichγ(tj) = xj . By the continuity of γ with respect to d0 and the fact d0 gener-ates the same topology as g(T ), we may select times t′j ∈ (tj , 1) which markthe first time after tj in which γ intersects ∂Bg(T )(xj , ϵ), i.e.,t′j = sup{t > tj : γ(s) ∈ Bg(T )(xj , ϵ) for all s ∈ (tj , t]}.Then since ϵ was chosen small enough so that (5.14) holds for y ∈ Bg(T )(xj , ϵ),we obtain∞ =∞∑j=1C−1ϵ11−2(n−1)α =∞∑j=1C−1[dg(T )(xj , γ(t′j))]11−2(n−1)α≤∞∑j=1d0(xj , γ(t′j)) ≤ Lengthd0(γ) <∞,128a contradiction. Thus (M∞, d0) is a complete metric space. The claimabout the pointed Gromov-Hausdorff convergence with the Fi being theapproximating functions is straightforward from (5.13). So it suffices toprove the claim.Proof of Claim 5.3.1. We will prove the inequalities in (5.12) individually.In what follows, we let C denote any large constant depending only onn,A, τ, v0, α (recall that α = C0σ also depends only on n,A, τ, v0) and mayvary line-by-line, and we will also drop the subscripts i for brevity. The right-hand side is identical to the proof of the analogous inequality in property(iv) of [5, Theorem 5.1], so we only give a sketch.By the α/t curvature bound and the Shrinking Balls Lemma we obtain,for 0 < s ≤ t ≤ T ,dg(0)(x, y) ≤ dg(s)(x, y) + C√s ≤(ts)(n−1)αdg(t)(x, y) + C√s. (5.15)If dg(t)(x, y) > 1 then the right-hand side of (5.12) is obvious, so assumedg(t)(x, y) ≤ 1 and define s ∈ (0, t] bys = t d21+2(n−1)αg(t) (x, y). (5.16)Then the right-hand side of (5.12) follows by plugging (5.16) directly into(5.15) and simplifying using the assumptions that t ≤ 1.Now we prove the left-hand side. Recall from the proof of Theorem Ethat we have some uniform lower bound of volume ratios up to some fixedscale along the flow (see in particular (5.9)). Thus by scaled versions ofLemma 5.1.3, we may obtain thatdg(t)(x, y) ≤ Cdg(0)(x, y) whenever√T̂−1t ≤ dg(0)(x, y) ≤132(5.17)where T̂ is from Lemma 5.1.3, and depends on n,A, τ, v0, α. If t ≥ T̂322 , thenthe α/t curvature bound allows us to conclude dg(t)(x, y) ≤ Cdg(0)(x, y)immediately. If√T̂−1t ≤ 132 ≤ dg(0)(x, y) ≤ 1, then we set x0 = x, x32 = yand choose intermediate points {xj}31j=1 satisfying31∑j=0dg(0)(xj , xj+1) ≤ 32dg(0)(x, y),√T̂−1t ≤ dg(0)(xj , xj+1) ≤ 132 .129Then we may conclude dg(t)(x, y) ≤ Cdg(0)(x, y) by applying (5.17) foreach j = 0, 1, . . . , 31, along with the triangle inequality. So it suffices toconsider x, y, t satisfying dg(0)(x, y) <√T̂−1t < 132 . In this case, write√s =√T̂ dg(0)(x, y) so that by the α/t curvature bound and (5.17), weobtaindg(t)(x, y) ≤ s−(n−1)αdg(s)(x, y)≤ Cs−(n−1)αdg(0)(x, y)≤ Cd1−2(n−1)αg(0) (x, y).This shows the left-hand side of (5.12) completes the proof of the claim5.3.2 Existence and properties of µOur technique to establishing the existence of the measure µ will beto show that the limits limt→0Volg(t)Ω and limi→∞VolgiFi(Ω) both exist and areequal whenever Ω is a precompact Borel set. In the case that Ω is not pre-compact, we can then define µ(Ω) as a limit of countably many precompactpieces. Before commencing the proof however, we give a reformulation ofour Volume Perturbation Lemma 5.1.1 in the context of Theorem E.Lemma 5.3.2. Suppose (Mn, g0) satisfies the hypotheses of Theorem E forsome choices of variables n ≥ 4, A, τ, v0 > 0, and write g(t), t ∈ [0, T ] to bethe corresponding complete Ricci flow. Given any precompact set Ω ⊂⊂ Msuch that Bg(0)(x, r) ⊂ Ω for some r ∈ (0, 18 ], then whenever 0 ≤ s ≤ t ≤min{ r2β2, T}, there holdsC−(t−s)Volg(s)Ω ≤ Volg(t)Ω ≤ C(t−s)Volg(s)Ω.Here C ≥ 1 is some constant that depends only on n,A, τ, r as well as thesmallest number N such thatΩ ⊂N⋃j=1Bg(0)(xj ,116).Proof. The proof follows from the properties of such a flow g(t) as laid outin (5.11) and a direct application of Lemma 5.1.1. In particular, we can130say as long as t ∈ [0, T ] satisfies t ≤ r2β2, or more generally t ≤ r24β2α(recallα ≤ 14(n−1)), we can apply the Shrinking Balls Lemma to see thatBg(t)(x, r/2) ⊂ Bg(0)(x, r) ⊂ Ω.Since r ≤ 18 , we have that r0 := r/2 ≤ 116 is a scale in which g(t) has alower bound of volume ratios v˜ from (5.11). Also, we can bound the integralscalar curvature on Ω as(∫Ω|R|n2 dVg(t)) 2n≤ N∑j=1∫Bg(0)(xj ,116)nn2 |Rm|n2 dVg(t) 2n≤ (nn2Nαn2 ) 2n = N 2nnα.Therefore, we can see that the result holds by applying Lemma 5.1.1 withr, σ there being r0, N2nnα respectively, and thus we may takeC = exp(N2nnαv˜−2n (r/2)−2)≤ exp(N2nnv˜−2n(n− 1)r2)where v˜ depends only on n,A, τ .We also require the following elementary lemma which will be appliedwith f(t) being either Volgi(t)Fi(Ω) or Volg(t)Ω in our proof.Lemma 5.3.3. Let f : (0, T ] → R be a positive continuous function satis-fyingC−(t−s)f(s) ≤ f(t) ≤ C(t−s)f(s)whenever 0 < s ≤ t ≤ T . Then for any ϵ > 0, there exists tϵ ∈ (0, T ] depend-ing only on ϵ as well as upper bounds of C and f(T ), such that whenever0 < s ≤ t ≤ tϵ, there holds|f(t)− f(s)| ≤ ϵ.In particular, limt→0f(t) exists.Proof. For any ϵ > 0, definetϵ = min{T,1logClog(1 +ϵCT f(T ))}.131Then whenever 0 < s ≤ t ≤ tϵ, we obtain|f(t)− f(s)| ≤ (C(t−s) − 1) supt′∈(0,T ]f(t′) ≤ (Ctϵ − 1)CT f(T ) ≤ ϵ.Armed with these preliminary lemmas, we are ready to prove the ex-istence and equality of the desired limits. We split our proof into threeclaims:1. First show that the time limit exists assuming that the interior of theset is not empty (Claim 5.3.4).2. Show that the two limits are equal assuming that the interior of theset is not empty (Claim 5.3.5).3. Remove the assumption about nonempty interior (Claim 5.3.6).Note that in the above steps, it is unambiguous whether the ‘interior’of a set is with respect to the metric d0 or the topology coming from thesmooth structure ofM∞ because we have already shown these are the same.Claim 5.3.4. For any precompact set Ω ⊂M∞ with nonempty interior, thelimitlimt→0Volg(t)Ωexists.Proof. Choose some p ∈M∞ and r ∈ (0, 18 ] such thatBd0(p, 2r) ⊂ Ω.For i large enough, we may insist thatBgi(0)(Fi(p), r) ⊂ Fi (Bd0(p, 2r)) ⊂ Fi(Ω).For this fixed choice of r > 0, we will denote TΩ = min{ r2β2 , T}. Now sinceΩ ⊂ M∞ is precompact, we may take finitely many points {xj}Nj=1 ⊂ M∞such thatΩ ⊂N⋃j=1Bg(TΩ)(xj ,1100).132Then restrict i ∈ N large enough so that for each 1 ≤ j ≤ N ,Fi(Bg(TΩ)(xj ,1100))⊂ Bgi(TΩ)(Fi(xj),150).Subsequently, by our definition of TΩ, our assumption of r ≤ 18 , and theShrinking Balls Lemma, we can say that for all i sufficiently large,Fi(Ω) ⊂N⋃j=1Bgi(Fi(xj),116).Then by Lemma 5.3.2, we can say that there exists a constant C > 0 de-pending on n,A, τ,N (but not on i) such that whenever 0 ≤ s ≤ t ≤ TΩ,C−(t−s)Volgi(s)Fi(Ω) ≤ Volgi(t)Fi(Ω) ≤ C(t−s)Volgi(s)Fi(Ω). (5.18)So for any fixed 0 < s ≤ t ≤ TΩ, we can take i → ∞, which shows that(5.18) transcends to the limit:C−(t−s)Volg(s)Ω ≤ Volg(t)Ω ≤ C(t−s)Volg(s)Ω whenever 0 < s ≤ t ≤ TΩ.We can therefore conclude that limt→0Volg(t)Ω exists by a direct application ofLemma 5.3.3.Claim 5.3.5. For any precompact set Ω ⊂ M∞ with nonempty interior,there holdslimi→∞VolgiFi(Ω) = limt→0Volg(t)Ω.Proof. We have already established that the limit limt→0Volg(t)Ω exists. Thusfor any ϵ > 0, we may assume t0 > 0 is small enough so that whenevers ∈ (0, t0],| limt→0Volg(t)Ω−Volg(s)Ω| < ϵ.Without loss of generality, we assume t0 ≤ TΩ where TΩ is defined in theproof of Claim 5.3.4. In what follows, we assume that i ≥ iΩ for some iΩ ∈ Nwhich may be increased several times. First, if we restrict iΩ large enoughso thatVolgi(t0)Fi(Ω) ≤ 2Volg(t0)Ω,then we obtain from (5.18) and Lemma 5.3.3 some small tϵ ∈ (0, t0] (here tϵdoes not depend on i as long as i ≥ iΩ) satisfying|Volgi(tϵ)Fi(Ω)−Volgi(0)Fi(Ω)| < ϵ.133Then we increase iΩ again if necessary to ensure that|Volg(tϵ)Ω−Volgi(tϵ)Fi(Ω)| < ϵ.A direct application of the triangle inequality completes the proof.Claim 5.3.6. For any precompact Borel set Ω ⊂ M∞ (no other assump-tions), there holdslimi→∞VolgiFi(Ω) = limt→0Volg(t)Ω.Proof. We begin by claiming thatlim inft→0Volg(t)Ω = lim supr→0limt→0Volg(t)(Ω ∪Bd0(x∞, r)). (5.19)The “≤” inequality is obvious, so assume for the sake of contradiction thatthere is some ϵ > 0 such thatϵ+ lim inft→0Volg(t)Ω ≤ lim supr→0limt→0Volg(t)(Ω ∪Bd0(x∞, r)). (5.20)Define r0 = n√ϵ2n+4v0, i.e., 2n+4v0rn0 = ϵ. Then select some r1 ∈ (0, r0) smallenough so thatlim supr→0limt→0Volg(t)(Ω ∪Bd0(x∞, r))≤ limt→0Volg(t)(Ω ∪Bd0(x∞, r1)) +ϵ4.(5.21)Since the limit on the right-hand side of (5.21) exists by Claim 5.3.5, wemay take t0 > 0 small enough so that whenever s ∈ (0, t0],limt→0Volg(t)(Ω ∪Bd0(x∞, r1)) ≤ Volg(s)(Ω ∪Bd0(x∞, r1)) +ϵ4. (5.22)Similarly, we can find a sequence tj → 0 such that for every j ∈ N,Volg(tj)Ω−ϵ4≤ lim inft→0Volg(t)Ω. (5.23)Thus whenever tj ≤ t0, we can see by (5.20) - (5.23) thatϵ4+ Volg(tj)Ω ≤ Volg(tj)(Ω ∪Bd0(x∞, r1))≤ Volg(tj)Ω+Volg(tj)(Bd0(x∞, r1)).(5.24)134We can assume that whenever i is large enough (depending on j) thatVolg(tj)(Bd0(x∞, r1)) ≤ 2Volgi(tj)Fi(Bd0(x∞, r1))≤ 2Volgi(tj)(Bgi(xi, 2r1)).Restricting j large enough so that tj ≤ (2r1)2β2, we obtain from Lemma 5.3.2and (5.24) thatϵ4≤ 2Volgi(tj)(Bgi(xi, 2r1)) ≤ 4Volgi(Bgi(xi, 2r1)).Here we have assumed that i is sufficiently large (depending on j), and thatj in turn is large enough to ensure that Ctj ≤ 2, where C is the constantfrom Lemma 5.3.2. Now a contradiction is immediately apparent becausewe have assumed r1 < r0, and we know that Volgi(Bgi(xi, 2r1)) ≤ v0(2r1)n.Thus we have proved that (5.19) holds. Moreover, using a similar (albeitless involved) approach to the proof of (5.19), we also obtainlim infi→∞VolgiFi(Ω) = lim supr→0limi→∞VolgiFi(Ω ∪Bd0(x∞, r)). (5.25)Not only can we say from (5.19) and (5.25) that the limits limi→∞VolgiFi(Ω)and limt→0Volg(t) exist, but we also can say that for any ϵ > 0, we can taker > 0 small enough so that| limi→∞VolgiFi(Ω)− limt→0Volg(t)Ω|≤ | limi→∞VolgiFi(Ω)− limi→∞VolgiFi(Ω ∪Bd0(x∞, r))|+ | limi→∞VolgiFi(Ω ∪Bd0(x∞, r))− limt→0Volg(t)(Ω ∪Bd0(x∞, r))|+ | limt→0Volg(t)(Ω ∪Bd0(x∞, r))− limt→0Volg(t)Ω| ≤ ϵ+ 0 + ϵ.This completes the proof of the claim.We are now ready to formally define the measure µ. For the sake ofbrevity, let us write Aj to denote the j’th annulus centered at x∞ withrespect to the metric d0. That is,Aj := Bd0(x∞, j) \Bd0(x∞, j − 1).Also we write Borel(Ω) to denote the σ-algebra of all Borel subsets of Ω andrecall a basic measure theory fact thatBorel(Aj) = {E ∩ Aj : E ∈ Borel(M∞)}.135Thus for each j ∈ N, we define µj : Borel(Aj)→ R byµj(E) = limt→0Volg(t)E (or equivalently, µj(E) = limi→∞VolgiFi(E)), (5.26)which is a well-defined function by Claim 5.3.6. Additionally, by an elemen-tary measure theory argument (I was only able to find a proof on Mathemat-ics Stack Exchange [37] and have replicated the argument at the end of thesection for the reader’s convenience, see Claim 5.3.7), we may conclude thateach µj is countably additive, or equivalently in this setting, that each tuple(Aj ,Borel(Aj), µj) is in fact a measure space. Now for any E ∈ Borel(M∞),we defineµ(E) :=∞∑j=1µj(E ∩ Aj).This can be seen to define a measure on M∞ because if {Ek}k∈N is somepairwise disjoint sequence of Borel sets, we write E =⋃k∈NEk and calculateµ(E) =∞∑j=1µj(E ∩ Aj) =∞∑j=1∞∑k=1µj(Ek ∩ Aj)=∞∑k=1∞∑j=1µj(Ek ∩ Aj) =∞∑k=1µ(Ek),where we have applied Tonelli’s Theorem and the countable additivity ofeach µj .Finally, to show that µ is Radon, we must show that it is outer and innerregular, i.e., thatµ(E) = sup{µ(K) : K ⊂ E is compact} = inf{µ(U) : U ⊃ E is open}for any E ∈ Borel(M∞). Since the proof of inner regularity is very similarto that of the outer regularity, we will only show the latter. To that end,fix some E ∈ Borel(M∞) and ϵ > 0. Just as in the proof of (5.19) in Claim5.3.6, for each j ∈ N we can find some xj ∈ M∞ and rj > 0 such thatBd0(xj , rj) ⊂ Aj and thatµ(Bd0(xj , rj)) ≤ϵ2j+2. (5.27)Write Bj = (E ∩ Aj) ∪Bd0(xj , rj), which is precompact in M∞, so we maychoose a finite sequence of points {yk}Nk=1 ⊂M∞ such thatBj ⊂N⋃k=1Bd0(yk,1100). (5.28)136Now choose t′j > 0 small enough such that whenever t ∈ (0, t′j ],|µj(Bj)−Volg(t)(Bj)| <ϵ2j+2. (5.29)Using the fact that Volg(t) is a Riemannian measure (and subsequently,Radon), we may choose some precompact open set Uj,t ⊂ M∞, contain-ing Bj , that satisfiesVolg(t)(Uj,t)−Volg(t)(Bj) <ϵ2j+2. (5.30)Moreover by intersecting Uj,t with the finite open cover from (5.28), we mayassumeUj,t ⊂N⋃k=1Bd0(yk,1100).Importantly, hereN is independent of t ∈ (0, t′j ]. Thus by a similar argumentto that of the proof of Claim 5.3.6, we obtain that|Volg(t)(Uj,t)− µ(Uj,t)| <ϵ2j+2(5.31)whenever t is sufficiently small.So for each j ∈ N, we select tj > 0 small enough so that all of (5.29),(5.30), (5.31) hold. Then withU =∞⋃j=1Uj,tj ,which clearly is open and contains E because each Uj,tj contains E ∩Aj , wemay apply (5.27) and (5.29) - (5.31), along with the fact that µ|Aj = µj , tosee thatµ(U)− µ(E)≤∞∑j=1µ(Uj,tj )−∞∑j=1µj(Bj) +∞∑j=1µj(Bd0(xj , rj))≤∞∑j=1[Volg(tj)(Uj,tj ) +ϵ2j+2]−∞∑j=1[Volg(tj)(Bj)−ϵ2j+2]+∞∑j=1ϵ2j+2≤∞∑j=1[ ϵ2j+2+ϵ2j+2+ϵ2j+2+ϵ2j+2]= ϵ.137This completes the proof.Before the conclusion of the section, we provide the following measuretheory fact that the limit of a sequence of measures is once again a measure.Claim 5.3.7 ([37] Limit of measures is a measure). Let (µj)j∈N be a sequenceof Borel measures on Ω such that µj(Ω) < ∞ for each j ∈ N and that thelimitlimj→∞µj(E)exists for every Borel set E ⊂ Ω. Then the limiting function µ(E) :=limj→∞µj(E) is itself a measure (in particular, µ is countably additive).Proof. Let {Ek}k∈N be a countable set of pairwise disjoint Borel sets. Weneed to show thatµ( ∞⋃k=1Ek)=∞∑k=1µ(Ek).Write E =⋃∞k=1Ek. The “≥” inequality is straightforward becauseµ(E) ≥ limi→∞µi(N⋃k=1Ek)=N∑k=1µ(Ek)holds for every N ∈ N, and thus we obtain µ(E) ≥ ∑∞k=1 µ(Ek). Thus itsuffices to prove the other inequality. We use a proof by contradiction andassume that the inequality is strict, so that we can say that there is someϵ0 > 0 such that for each n0 ∈ N there exits infinitely many m ∈ N with∞∑k=n0µm(Ek) ≥ 2ϵ0. (5.32)If it is not clear why we may assume (5.32), here is the reasoning: If thisweren’t true, then for every ϵ > 0 there would exist n0,m0 ∈ N such that∑∞n=n0µm(En) < ϵ for all m > m0. But then, for m > m0 sufficiently large,138we would have∞∑n=1µ(En) ≥n0−1∑n=1µ(En) = limm′→∞n0−1∑n=1µm′(En)≥n0−1∑n=1µm(En)− ϵ ≥∞∑n=1µm(En)− 2ϵ= µm( ∞⋃n=1En)− 2ϵ m→∞−→ µ( ∞⋃n=1En)− 2ϵ,i.e.,∑n µ(An) ≥ µ(⋃nEn) which we assumed was not the case.Using (5.32), we construct two increasing sequences mk, Nk recursivelyas follows: Choose m1 such that∑∞k=1 µm1(Ek) ≥ 2ϵ0 and then chooseN1 sufficiently large so that∑∞k=N1+1µm1(Ek) ≤ ϵ0/5, which in particularimplies∑N1k=1 µm1(Ek) ≥ ϵ0. Now having constructed mn and Nn, choosemn+1 so that|µmn+1(Ek)− µ(Ek)| ≤2−(k+1)ϵ05for all k ≤ Nn,and ∞∑k=Nn+1µmn+1(Ek) ≥ 2ϵ0.Then choose Nn+1 > Nn such that∞∑k=Nn+1+1µmn+1(Ek) ≤ϵ05.Note again, that this implies∑Nn+1k=Nn+1µmn+1(Ek) ≥ ϵ0. Now defineB :=⋃n∈NNn+1⋃k=Nn+1EkGiven any n ∈ N, we have139|µmn+1(B)− µmn(B)| =∣∣∣∣∣∣∑l∈NNl+1∑k=Nl+1(µmn+1(Ek)− µmn(Ek))∣∣∣∣∣∣≥∣∣∣∣∣∣Nn+1∑k=Nn+1µmn+1(Ek)∣∣∣∣∣∣−∣∣∣∣∣∣Nn+1∑k=Nn+1µmn(Ek)∣∣∣∣∣∣−∣∣∣∣∣∣∞∑k=Nn+1+1µmn+1(Ek)∣∣∣∣∣∣−∣∣∣∣∣∣∞∑k=Nn+1+1µmn(Ek)∣∣∣∣∣∣−Nn−1∑k=1∣∣µmn+1(Ek)− µmn(Ek)∣∣︸ ︷︷ ︸≤2−kϵ0/5≥ ϵ0 − ϵ05− ϵ05− ϵ05− ϵ05=ϵ05Therefore (µmn(B))n∈N is not a Cauchy sequence, contradicting the factthat µi(B) converges. This completes the proof.5.4 Topological and volume rigidityIn this section, we will prove Theorem G, which states that whenever amanifold has small enough (global) curvature concentration with respect tothe inverse of its (global) Sobolev constant, then it is topologically rigid. Inparticular, we will see that it must be homeomorphic to Rn, or even stronger,diffeomorphic for n > 4. In this sense, Theorem G is a generalization ofTheorem A although we also require an additional assumption about theupper bound on volume ratios. Moreover, we are able to conclude rigidvolume bounds at infinity and - if you’re willing to assume the curvatureconcentration is even smaller - global volume rigidity.We begin by recalling the precise statement of Theorem G.Theorem. For all n ≥ 4, A, v0 > 0, there exists σ > 0 depending on n,A, v0such that the following holds. For any ϵ ≤ σ, we denote Mϵ to be the familyof complete noncompact (not necessarily bounded curvature) Riemannianmanifolds (Mn, g) satisfying:140(i) VolgBg(x, r) ≤ v0rn for all x ∈M, r > 0;(ii) ν¯(M, g) ≥ −A;(iii)(∫M |Rm|n2 dVg) 2n ≤ ϵ; and(iv) infx∈MRcg(x) > −∞.Then each of the following rigidity properties holds:(a) If (Mn, g) ∈Mσ, then Mn is homeomorphic to Rn, and if n ̸= 4 thenthe homeomorphism can be taken to be a diffeomorphism.(b) For any (M, g) ∈ Mσ, the volume growth of (M, g) at infinity is atleast that of Euclidean space, uniformly with respect to the base point.More precisely,lim infr→∞,x∈MVolgBg(x, r)rn≥ ωn.(c) By restricting ϵ smaller, we may assume that all volume ratios areuniformly close (from below) to that of Euclidean space, uniformlywith respect to the manifold, base point, and scale. More precisely,lim infϵ→0{VolgBg(x, r)rnwhere (M, g) ∈Mϵ, x ∈M, r > 0}≥ ωn.To setup the proof, we begin by assuming that σ(n,A, v0) is boundedfrom above by σ(n,A, 1, v0) from the statement of Theorem E (in otherwords, set τ = 1), and we will further shrink σ if necessary in the proofs ofproperties (a) and (b). Because the assumptions of the theorem are scale-invariant, we may apply Theorem E to successively extreme blow-downsR−2g, which (after scaling back) gives a family of Ricci flows gR(t) onM × [0, TR2], for R arbitrarily large. Note in particular that we have usedproperties of the local ν¯-functional:ν¯(BR−2g(x, 1), R−2g, 1) = ν¯(Bg(x,R), g, R2) ≥ ν¯(M, g) ≥ −A.Then just as in the proof of Theorem E, we can extract a converging subse-quencegRj (t)C∞loc(M×[0,∞))−−−−−−−−−→ g(t),141where g(0) = g. By virtue of the conclusions (5.4) (that is, the conclusionsof Theorem E) being rescaled as well, we can conclude that the long-timecomplete flow g(t) satisfies the following for all t ∈ (0,∞):|Rm|g(t) ≤ C0σt−1;injg(t) ≥ C−10√t; and(∫M |Rm|n2 dVg(t)) 2n ≤ C0σ.(5.33)Here C0 ≥ 1 depends only on n,A, v0 (recall we fixed τ = 1 here). Moreover,by the proof of Theorem E, we can also conclude that we have a global L2-Sobolev inequality and a uniform lower bound on all volume ratios alongthe flow. More precisely, we know that (5.8) holds for the metrics R−2g. Soas R→∞, we obtainSob(M, g(t)) ≤ [CS , 0] (5.34)for all t ∈ [0,∞), where CS = CS(n,A, 1) is independent of t ∈ [0,∞).Similarly, by rescalings of (5.9), we obtaininfx∈M,r>0,t≥0Volg(t)Bg(t)(x, r)rn≥ v˜(n,CS). (5.35)Now with these properties we are ready to show that conclusions (a), (b),(c) hold.Proposition 5.4.1. Conclusion (a) holds.Proof. The proof is similar to the proof of [5, Theorem 1.2]. First note thatsince assumption (i) as well as each of the conclusions in (5.33) and (5.35)are scale-invariant, the flows gR(t) = R−2g(R2t) satisfy these conditions aswell. Lemma 5.1.3 then implies that whenever t ∈ [0, T̂ (n,C0σ, v˜, C0σ)], wehaveBgR(0)(x0, 1) ⊂ BgR(t)(x0, 22n+2v0v˜−1).Fix some T > T̂ large enough so that C−10√T ≥ 22n+3v0v˜−1, then shrinkσ accordingly so that for any Ricci flow h(t), t ∈ [0, T ] satisfying (5.33) (inparticular, the gR(t) satisfy this for any large R),dh(T )(x0, ·) ≤ 2dh(T̂ )(x0, ·).Scaling back, we can say that Bg(0)(x0, R) is diffeomorphic to Rn for allR > 0. The fact that M is homeomorphic to Rn follows by [3], and the142homeomorphism can be taken to be a diffeomorphism when n ̸= 4 becauseR4 is only Euclidean space with exotic differential structures.Proposition 5.4.2. Conclusion (b) holds.Proof. The proof is similar to the proof of Theorem C (from Section 4.1).First, we further shrink σ if necessary to ensure thatσ ≤ min{δ1C0CS,14β2C0}where δ1 = δ(n) is the dimensional constant from Theorem A. We maynow consider σ to be fixed. Then by the smallness of the global curvatureconcentration in (5.33) and the L2-Sobolev inequality (5.34), we may applyTheorem A to the bounded curvature metric g(1) to conclude faster than1/t curvature decay:limt→∞ t supx∈M|Rm|g(t)(x) = 0. (5.36)Now we argue by contradiction by assuming there is some δ > 0 andsequences xj ∈M and Rj →∞ withVolgBg(xj , Rj) < (1− δ)ωnRnj . (5.37)Consider the blow-down pointed flows (M, gj(t), xj), t ∈ (0, 1], where gj(t) =R−2j g(R2j t). By the estimates (5.33), we obtain - via Hamilton’s Compact-ness [23] - that gj(t) subconverges to a limiting complete flow (M∞, g∞(t)),t ∈ (0, 1]. Furthermore, by nature of the uniform faster than 1/t curvaturedecay (5.36) and uniform lower bound on volume ratios (5.35), we can seethat this flow must be isometric to the static flow on flat Euclidean space.Now take some tδ small enough (depending on δ, C0, σ, v˜ and thus only onδ, n,A, v0) such thatmin{exp(−4nC0σv˜− 2n tδ),(1− β√C0σtδ)n} ≥ (1− δ) 13 . (5.38)Then we can say that for j sufficiently large, we haveVolgj(0)Bgj(0)(xj , 1) ≥ (1− δ)13Volgj(tδ)Bgj(0)(xj , 1)≥ (1− δ) 13Volgj(tδ)Bgj(tδ)(xj , 1− β√C0σtδ)≥ (1− δ) 23(1− β√C0σtδ)nωn≥ (1− δ)ωn.143Here we have used Lemma 5.1.1 along with |Rg| ≤ n|Rm|g (see (1.19)) andthe first condition on tδ in (5.38) in the first line, the Shrinking Balls Lemmain the second line, the pointed convergence (M, gj(tδ), xj) → (Rn, gE , 0) inthe third line, and the second condition on tδ in (5.38) in the final line. Afterscaling back, we can see immediately that this contradicts our assumption(5.37), whenever Rj is sufficiently large.Remark 5.4.3. Precisely how large R needs to be to derive a contradictionin the preceding proof not only depends on the variables δ, n,A, v0, but alsoon the rate of the convergence in (5.36), which in turn possibly depends onsome other properties of g.Proposition 5.4.4. Conclusion (c) holds.Proof. We continue using the notation of Mϵ as in the theorem statementwhere n,A, v0 are all fixed. We first have the following claim:Claim 5.4.5. For any δ > 0, we can take ϵ small enough (depending onlyon δ, n,A, v0), so that for any (M, g) ∈ Mϵ and associated long-time Ricciflow g(t) as given above, there holdsVolg(t)Bg(t)(x,√t) ≥ (1− δ)ωntn2 (5.39)for all (x, t) ∈M × (0,∞).Proof of Claim. Since the hypotheses and conclusion of the claim are in-variant under scaling, it suffices to prove (5.39) holds only for some shorttime interval t ∈ [0, T1]. Then since we are working locally, we may assumethat the flow we are working on is actually complete with bounded curvatureand satisfying the estimates (5.4). This assumption is validated by the factthat our (possibly initially unbounded curvature) Ricci flow is obtained by asmooth local limit of such flows (recall the proof of Theorem E). The proofis then identical to the beginning of the proof of [4, Theorem 4.2].The remainder of the proof is similar to the proof of part (b) with thenotable difference being that instead of choosing a small time, we make ϵsmall instead. Along those lines, given δ > 0, let ϵδ/2 be as given in Claim5.4.5. Again by the invariance under scaling, we may assume that r = 1.First assume that ϵ is small enough so thatϵ ≤ min{ϵδ/2,14β2C0}.144Then writing tϵ =(1− β√C0ϵ)2< 1, we apply Lemma 5.1.1, together withthe Shrinking Balls Lemma and Claim 5.4.5 to see that(1− δ/2)ωntn2ϵ ≤ Volg(tϵ)Bg(tϵ)(x,√tϵ)≤ Volg(tϵ)Bg(0)(x, 1)≤ exp(4nC0ϵv˜− 2n tϵ)Volg(0)Bg(0)(x, 1).The result follows if we further shrink ϵ if necessary to ensure that(1− δ/2)(1− β√C0ϵ)nexp(−4nC0ϵv˜− 2n)≥ 1− δ.145Bibliography[1] Richard Bamler. 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Ricci flow : curvature concentration and Sobolev bounds
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