UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The Kähler-Ricci flow on non-compact manifolds Li, Ka-Fai 2018

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2018_september_li_kafai.pdf [ 577.74kB ]
Metadata
JSON: 24-1.0367001.json
JSON-LD: 24-1.0367001-ld.json
RDF/XML (Pretty): 24-1.0367001-rdf.xml
RDF/JSON: 24-1.0367001-rdf.json
Turtle: 24-1.0367001-turtle.txt
N-Triples: 24-1.0367001-rdf-ntriples.txt
Original Record: 24-1.0367001-source.json
Full Text
24-1.0367001-fulltext.txt
Citation
24-1.0367001.ris

Full Text

The Ka¨hler-Ricci flow on non-compact manifoldsbyKa-Fai LiA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)May 2018c© Ka-Fai Li, 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:The Ka¨hler-Ricci flow on non-compact manifoldssubmitted by Ka-Fai Li in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin MathematicsExamining Committee:Albert ChauSupervisorJingyi ChenSupervisory Committee MemberAilana FraserSupervisory Committee MemberYoung-Heon KimUniversity ExaminerJoanna KarczmarekUniversity ExamineriiAbstractWe first study the general theory of Ka¨hler-Ricci flow on non-compact complexmanifolds. By using a parabolic Schwarz lemma and a local scalar curvature es-timate, we prove a general existence theorem for Ka¨hler metrics lying in the C∞loc-closure of complete bounded curvature Ka¨hler metrics that are uniformly equiva-lent to a fixed background metric. In particular we do not assume any curvaturebounds. Next, we compare the maximal existence time of two complete boundedcurvature solutions by using the equivalence of the initial metrics and using this,we also estimate the maximal existence time of a complete bounded curvature so-lution in terms of the curvature bound of a background metric. We also prove auniqueness theorem for Ka¨hler-Ricci flow which slightly improves the result of[13] in the Ka¨hler case.We apply the above results to study the Ka¨hler-Ricci flow on some specificnon-compact complex manifolds. We first study the Ka¨hler-Ricci flow on Cn. Byapplying our general existence theorem and existence time estimate, we show thatany complete non-negatively curved U(n)-invariant Ka¨hler metric admits a long-time U(n)-invariant solution to the Ka¨hler-Ricci flow, and the solution convergesto the standard Euclidean metric after rescaling.Then we study the Ka¨hler-Ricci flow on a quasi-projective manifold M \D. Bymodifying the approximation theorem of [1] and applying a general existence the-orem of Lott-Zhang [23], we construct a Ka¨hler-Ricci flow solution starting fromcertain smooth Ka¨hler metrics. In particular, if the metric is the restriction of asmooth Ka¨hler metric in the ambient space M, then the solution instantaneouslybecomes complete and has cusp singularity at D. We also produce a solution start-ing from some complete metrics that may not have bounded curvature, and theiiisolution is likewise complete with cusp singularity for positive time. On the otherhand, if the initial data has bounded curvature and is asymptotic to the standardcusp model at D in a certain sense, we find the maximal existence time of thecorresponding complete bounded curvature solution to the Ka¨hler-Ricci flow.ivLay SummaryThe Ricci flow was introduced by Richard Hamilton in 1982 to solve the famousPoincare´ conjecture in mathematics, and it is a geometric process that deforms theshape of a given space according to how the space is curved. Hamilton provedthat some nicely curved 3 dimensional spaces deform to a sphere along the Ricciflow. Similar results have been proved for the Ricci flow since then. On the otherhand, some spaces cannot be deformed at all due to their roughness, and this thesisdiscusses conditions under which a space can be deformed along the Ricci flow,and when is there a unique way to carry out the deformation. We also apply theRicci flow to spaces with certain symmetries. For example, we prove that certainrotationally symmetric spaces will be flattened along the Ricci flow, and we relatethis to a well known conjecture in geometry.vPrefaceAll of the work presented in this thesis was conducted as I was a PhD student inthe Mathematics Department of University of British Columbia.Materials in Chapter 2 and 3 are from [10], [11] and [12], these are joint workswith Professor Albert Chau and Professor Luen-Fai Tam and they are all published.In [10], I contributed the parabolic Schwarz computations and the construction ofbackground U(n)-invariant metrics. In [11], all authors contributed equally. In[12], I made contributions to the existence of long-time solution to U(n)-invariantKa¨hler-Ricci flow. Some arguments in [12] are replaced by new original argumentsin this thesis.Materials in Chapter 4 are from [9], which is a joint work with Professor AlbertChau and Doctor Liangming Shen, it has been posted on arXiv. In this work, allauthors contributed equally.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Non-negatively curved U(n)-invariant metrics . . . . . . . . . . . 51.3 Ka¨hler-Ricci flow on quasi-projective manifolds . . . . . . . . . . 62 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 C0 estimates in terms of background metrics . . . . . . . . . . . . 112.3 General existence theorems . . . . . . . . . . . . . . . . . . . . . 182.4 A uniqueness theorem and an existence time estimate . . . . . . . 223 U(n)-invariant Ka¨hler metrics . . . . . . . . . . . . . . . . . . . . . 273.1 Background materials . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Short time existence of U(n)-invariant Ka¨hler-Ricci flow . . . . . 313.3 Long time solution . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Bounding the scalar curvature . . . . . . . . . . . . . . . . . . . 43vii3.5 Convergence after rescaling . . . . . . . . . . . . . . . . . . . . . 484 Quasi-projective manifolds . . . . . . . . . . . . . . . . . . . . . . . 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . 554.2.1 Proof of Theorem 4.1.1 when ϕ0 ∈C∞(M) . . . . . . . . 564.2.2 Proof of Theorem 4.1.1 when ϕ0 ∈ L∞(M)⋂C∞(M) . . . 624.3 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 Approximate solutions ωα, j(t) . . . . . . . . . . . . . . . 654.3.2 A priori estimates for ωα, j(t) . . . . . . . . . . . . . . . . 674.3.3 Completion of Proof of Theorem 4.1.2 . . . . . . . . . . . 734.4 Proof of Theorem 4.1.3 . . . . . . . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.1 Interior estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.2 Plurisubharmonic functions . . . . . . . . . . . . . . . . . . . . . 83viiiAcknowledgmentsI would like to thank my supervisor Professor Albert Chau, not only for his guid-ance and motivation during my Ph.D. study, but also for his support during mytough times. I would also like to thank Professor Jingyi Chen and Professor AilanaFraser for serving on my thesis committee. I am also grateful to Liangming Shenand John Ma for many helpful discussions. Last but not least, I would like to thankmy family and my friends outside of UBC for their encouragement and support.ixChapter 1IntroductionIn this thesis, we study the Ka¨hler-Ricci flow on non-compact complex manifolds.Let (M,g) be a n dimensional Ka¨hler manifold, a smooth family of Ka¨hler metricsg(t) is said to satisfy the Ka¨hler-Ricci flow on M× [0,T ) starting from g if∂∂ tgi j¯(t) =−Ri j¯(g(t))g(0) = g,(1.0.1)on M× [0,T ), where Ri j¯(g(t)) is the Ricci curvature of g(t). Any Ka¨hler metriccan also be represented by a Ka¨hler form, if ω is the Ka¨hler form of g then theequation (1.0.1) is equivalent to∂∂ tω(t) =−Ric(ω(t))ω(0) = ω,(1.0.2)where Ric(ω(t)) = −√−1∂ ∂¯ logωn(t) is the Ricci form of ω(t). In the follow-ing content, we will use Greek alphabets to denote a Ka¨hler form while Romanalphabets will be used to denote a Ka¨hler metric.The thesis is divided into three parts. In the first part, we study the generaltheory of Ka¨hler-Ricci flow on non-compact complex manifolds. We prove anexistence theorem for initial metrics that can be approximated by complete Ka¨hlermetrics of bounded curvature satisfying some C0 conditions. We also prove anexistence time estimate and a uniqueness theorem for complete bounded curvature1solutions.In the second part, we study the Ka¨hler-Ricci flow on Cn starting from a com-plete U(n)-invariant Ka¨hler metric. We show that if the U(n)-invariant Ka¨hlermetric has non-negative holomorphic bisectional curvature, then the Ka¨hler-Ricciflow has a corresponding long-time bounded curvature solution to the Ka¨hler-Ricciflow. And the solution we obtain converges to the standard Euclidean metric afterrescaling.In the third part, we study the Ka¨hler-Ricci flow on quasi-projective manifold.A Ka¨hler manifold M is called a quasi-projective manifold if M = M \D, whereM is compact Ka¨hler and D ⊂ M is a divisor with normal crossings. We willconstruct a solution to Ka¨hler-Ricci flow starting from certain initial metrics on Mand the solution we construct becomes instantaneously complete and with a cuspsingularity at D. We also estimate the existence time of our solution.1.1 General TheoremsOne of the most fundamental questions in the study of Ka¨hler-Ricci flow is theexistence of solutions. Unlike the Ka¨hler-Ricci flow on compact Ka¨hler manifolds,the existence problem on non-compact Ka¨hler manifolds is not a direct conse-quence of general theory. The most classical result which is due to Shi( [26, 27])states that if (M,g) is a complete Ka¨hler manifold with bounded curvature, thenthere is a complete bounded curvature solution to (1.0.1). Shi’s solution has beenused extensively in the application of Ricci flow on non-compact Riemannianmanifolds. Chen-Zhu [13] showed that Shi’s solution is unique among completebounded curvature solutions while Lott-Zhang [23] gave an analytic characteriza-tion of the maximal existence time of the solution. As a result, given a completebounded curvature solution, we can discuss the maximal existence time of the cor-responding complete bounded curvature solution to Ka¨hler-Ricci flow. There areother more general existence results concerning complete Ka¨hler metrics on non-compact manifolds in the literature, but they often require rather specific curvatureassumptions and their uniqueness and existence time are often not known.In case the initial metric has unbounded curvature, we prove that if it can be ap-proximated by certain complete bounded curvature metrics, then we can construct2a solution to 1.0.1:Theorem 1.1.1. Let g0 be a complete continuous Hermitian metric on a non-compact complex manifold Mn. Suppose there exists a sequence {hk,0} of smoothcomplete Ka¨hler metrics with bounded curvature on M converging uniformly oncompact subsets to g0 and another complete Ka¨hler metric gˆ on M with boundedcurvature and holomorphic sectional curvature bounded from above by K ≥ 0 suchthat(i) 1C gˆ≤ hk,0 ≤Cgˆ for some C independent of k;(ii) hk has bounded curvature for every k.Let T = 1/(2nCK) if K > 0, otherwise let T = ∞. Then the Ka¨hler-Ricci flow(1.0.1) has a smooth solution g(t) on M× (0,T ) such that(a) (1/(nC)−2Kt)gˆ≤ g(t)≤ B(t)gˆ on M× (0,T ) for some positive continuousfunction B(t) depending only on C, gˆ and n.(b) g(t) has bounded curvature for t > 0. More precisely, for any 0 < T ′ < Tand for any l ≥ 0 there exists a constant Cl depending only on C, l, T ′, gˆ andthe dimension n such thatsupM|∇lRm(g(t))|2g(t) ≤Clt l+2,(c) g(t) converges uniformly on compact subsets to g0 as t→ 0.Moreover, if g0 is smooth and {hk,0} converges smoothly and uniformly oncompact subsets of M, then g(t) extends to a smooth solution on M× [0,T ) withg(0) = g0.The proof of this theorem is based on estimating the local equivalence betweenthe evolving metric and the fixed background metric gˆ. This is achieved by using aparabolic Schwarz lemma, an Aubin-Yau trace estimate and a local scalar curvatureestimate. After establishing the equivalence, the higher order estimates follow fromEvans-Krylov theory. Furthermore, our proof to this theorem actually shows that ifh is a complete bounded curvature Ka¨hler metric such that 1C gˆ≤ h≤Cgˆ for some3constant C, then the maximal existence time of Shi’s solution satisfies T ≥ 12nCKif K > 0 and T = ∞ otherwise. That is, we can estimate the existence time ofa complete bounded curvature solution by the curvature bound of an equivalentcomplete bounded curvature Ka¨hler metric.On the other hand, we will also prove a theorem which compares the maximalexistence time of two complete bounded curvature metrics using their C0 data.Theorem 1.1.2. If g and h are two equivalent complete Ka¨hler metrics of boundedcurvature and C is a constant such that g ≤ Ch, then T (g) ≤ CT (h). Here T (·)is the maximal existence time of the complete bounded curvature solution to theKa¨hler-Ricci flow starting from the given metric.In particular, our result implies that if g is a complete Ka¨hler metric of boundedcurvature with T (g) = ∞, then this property is shared by all complete boundedcurvature Ka¨hler metrics equivalent to g.We also prove a theorem about the uniqueness:Theorem 1.1.3. Let (Mn, ĝ) be a complete non-compact Ka¨hler manifold. Supposethere is an exhaustion function ζ > 0 on (Mn, ĝ) with limx→∞ ζ (x) = ∞ such that|∂ ∂¯ζ |ĝ and |∇̂ζ |ĝ are bounded.Let g1(x, t) and g2(x, t) be two solutions of the Ka¨hler-Ricci flow (1.0.1) onM× [0,T ] with the same initial data g0(x) = g1(x,0) = g2(x,0). Suppose there isa positive function σ with limx→∞ logσ(x)/ logζ (x) = 0 such that the followingconditions hold for all (x, t) ∈M× [0,T ]:(i)ĝ(x)≤ ζ (x)g1(x, t); ĝ(x)≤ ζ (x)g2(x, t),(ii)−σ(x)≤ det((g1)i j¯(x, t))det((g2)i j¯(x, t))≤ σ(x).Then g1 ≡ g2 on M× [0,T ].This theorem implies that if g1 and g2 are uniformly equivalent to ĝ on M×[0,T ], then g1 ≡ g2. This slightly improves the uniqueness result of Chen-Zhu inthe Ka¨hler case.41.2 Non-negatively curved U(n)-invariant metricsThe result in Chapter 3 is related to Yau’s uniformization conjecture for non-compact Ka¨hler manifolds: Let (Mn,g) be a complete non-compact Ka¨hler man-ifold with positive holomorphic bisectional curvature, then M is biholomorphicto Cn. This is a long standing conjecture and there have been several partial re-sults so far, in particular using the Ka¨hler-Ricci flow. For example, Chau-Tam[8] used Ka¨hler-Ricci flow to prove that a complete non-compact Ka¨hler manifoldwith bounded non-negative holomorphic bisectional curvature and maximal vol-ume growth is biholomorphic to Cn. Recently, this result was generalized by Liu[21] by removing the bounded curvature condition. Liu did not use Ka¨hler-Ricciflow in his proof, but his result was re-proved by Lee-Tam [20] using the Ka¨hler-Ricci flow.The U(n)-invariant Ka¨hler metrics on Cn were first studied by Wu-Zheng [33]in an attempt to give examples of positively curved compete Ka¨hler metrics on Cn.Prior to their works, there were only three examples of positively curved completeKa¨hler metrics on Cn and they are all U(n)-invariant ( [6, 7, 19]). In Wu-Zheng’swork, they gave a very convenient parametrization of the complete U(n)-invariantmetrics with non-negative bisectional curvature, in particular illustrating that thesegenerically had unbounded curvature. Motivated by these examples, Yang-Zheng[35] showed that if g(t) is a complete U(n)-invariant solution to the Ka¨hler-Ricciflow with initial metric having non-negative holormophic bisectional curvature,then the solution will also have non-negative holomorphic bisectional curvaturefor all time. Furthermore the asymptotic volume ratio remains constant along theflow. They also used the construction of Cabezas-Rivas and Wilking [2] to producea short time solution for complete U(n)-invariant initial metric with non-negativesectional curvature satisfying some technical assumptions. Using a different ap-proach, we prove that:Theorem 1.2.1. Let g0 be a complete U(n)-invariant Ka¨hler metric on Cn withnon-negative holomorphic bisectional curvature. Then(i) the Ka¨hler-Ricci flow (1.0.1) has a unique smooth long-time U(n)-invariantsolution g(t) which is equivalent to g0 and has bounded non-negative bisec-tional curvature;5(ii) g(t) converges, after rescaling at the origin, to the standard Euclidean metricon Cn.Here g(t) is equivalent to g0 means that for all T < ∞, there exists constantC such that 1C g0 ≤ g(t) ≤ Cg0 for all t ∈ [0,T ]. Our result does not assume anyvolume growth rate on the initial metrics but yet the solution converges to thestandard Euclidean metric in a certain sense. This showed that the Ka¨hler-Ricciflow may still be used to attack Yau’s conjecture even if we do not assume anyvolume growth conditions.1.3 Ka¨hler-Ricci flow on quasi-projective manifoldsIn Chapter 4, we study the Ka¨hler-Ricci flow on quasi-projective manifolds. AKa¨hler manifold M is called a quasi-projective manifold if M = M \D, where M isKa¨hler and D⊂M is a divisor with normal crossings. M is clearly a non-compactmanifold, given any Ka¨hler metric on M, we can define different notions of singu-larity at D. A metric on M is said to have cusp singularities at D if it is equivalent toa Carlson-Griffiths type form η− i∂ ∂¯ log log2 |S|2 where η is a Ka¨hler form on Mand S is the holomorphic section of [D] that vanishes on D. The Carlson-Griffithsform is analogous to the Poincar´e metric on the punctured disk, with the divisorcorresponding to the origin. In particular, cusp metrics are complete on M withpossibly unbounded curvature. In Lott-Zhang [23], they considered bounded cur-vature cusp metrics satisfying certain asymptotics conditions at D, and showed thattheir maximal existence time for (1.0.1) is bounded above byT[ω0] := sup{T : [η ]+T (c1(KM)+ c1(OD)) ∈KM} (1.3.1)whereKM is the Ka¨hler cone of M.The main theorems in Chapter 4 are:Theorem 1.3.1. Let ω0 = η+ i∂ ∂¯ϕ0 be a smooth Ka¨hler metric on M, and let T[ω0]be given by (1.3.1).(a) If ϕ0 ∈ L∞(M)⋂C∞(M)⋂PSH(M,η) and ω0 ≥ cη for some constant c > 0.6Then (1.0.1) has a unique smooth solution ω(t) on M× [0,T[ω0]) wherec1(t)ωˆ ≤ ωt ≤ c2(t)ωˆ (1.3.2)for all t ∈ (0,T[ω0]) and some positive functions ci(t).(b) Let ϕ0 ∈ C∞(M)⋂PSH(M,η) have zero Lelong number and ω0 ≥ cωˆ forsome c > 0. Then the Ka¨hler -Ricci flow (1.0.1) has a smooth solution ω(t)on M× [0,T[ω0]) andω(t)≥ (1n− 4Kˆtc)ωˆ (1.3.3)for all t ≤ c4nKˆ.As a particular case of part (a), if ω0 is the restriction of a Ka¨hler metric onM or has conical singularities at D, the solution instantaneously becomes completeand is equivalent to a cusp metric for t > 0. On the other hand, the initial metricin (b) is complete with possibly unbounded curvature, and there the solution islikewise complete for a uniform amount of time.There have been several earlier works about solutions to the Ka¨hler-Ricci flowwith conical singularities at D (see for example [14], [22], [25]). These worksestablished the characterization of the existence time in terms of the cohomologyof M, and also produced an instantaneously conical solution starting from initialdata that is restriction from smooth metrics on M. Our results can be viewed as acuspidal versions of these results.In addition to Theorem 1.3.1, we will also discuss the maximal existence timeof a complete bounded curvature solution starting from a Ka¨hler metric which hasbounded curvature and is asymptotic to a cusp model in a certain sense.7Chapter 2General Theorems2.1 BackgroundThe Ricci flow was first introduced by Hamilton [18] to prove that a closed 3 di-mensional Riemannian manifold with positive Ricci curvature is diffeomorphic toa spherical space form or its quotient. Since then the Ricci flow has been appliedto many problems in geometry, it was first applied to Ka¨hler manifolds by Cao in[4] who used the flow to re-prove the Calabi-Yau theorem.The behavior of Ricci flow on compact Ka¨hler manifolds has been well studied.Suppose that (M,g) is a closed Ka¨hler manifold, then it is known that a short-timesolution g(t) to the Ricci flow exists and the solution remains Ka¨hler . The exis-tence follows from the fact that on Ka¨hler manifolds the Ricci flow is a parabolicsystem of partial differential equations. In fact by applying the ∂ ∂¯ -lemma, it canbe shown here the Ricci flow is equivalent to a parabolic Monge-Ampe`re equa-tion for a scalar function. With the aid of this equivalence, the uniqueness followsreadily by a maximum principle argument. The maximal existence time of the so-lution was characterized by Tian-Zhang [31] in terms of the cohomology class ofthe initial metric:Theorem 2.1.1 (Tian-Zhang [31]). Let (M,ω) be a closed Ka¨hler manifold, sup-pose that ω(t) solves the Ka¨hler-Ricci flow. Then the maximal existence time of the8solution is given byT[ω] = sup{t > 0 : [ω]− tc1(M)> 0}.Here we say a cohomology class [α] > 0 if there is a Ka¨hler form inside the class[α], and c1(M) is the first Chern class of the manifold, which is equal to the class[Ric(ω)] for arbitrary Ka¨hler metric ω on M.When the Ka¨hler manifold (M,g) is non-compact, we do not expect the Ricciflow to have a solution unless g satisfies some extra conditions. And even if itexists, the solution g(t) may not be Ka¨hler or even Hermitian for t > 0. In caseg(t) solves the Ricci flow: ∂∂ tg(t) =−Ric(g(t))g(0) = g.(1.0.1)and g(t) is Ka¨hler for all t, we will call it a solution to Ka¨hler-Ricci flow. Theresearch of Ka¨hler-Ricci flow on non-compact manifolds was pioneered by Shi([26, 27]), and he proved the short time existence of Ka¨hler-Ricci flow on completenon-compact Ka¨hler manifolds with bounded curvature:Theorem 2.1.2. Let (Mn,g0) be a complete non-compact Ka¨hler manifold withcurvature bounded by a constant K. Then for some 0 < T ≤ ∞ depending only onK and the dimension n, there exists a smooth solution g(t) to (1.0.1) on M× [0,T )with g(0) = g0 such that(i) g(t) is Ka¨hler and equivalent to g0 for all t ∈ [0,T );(ii) g(t) has uniformly bounded curvature on M× [0,T ′] for all 0< T ′< T . Moreprecisely, for any l ≥ 0 there exists a constant Cl depending only on l, T ′, g0and the dimension n such thatsupM|∇lRm(g(t))|2g(t) ≤Clt l,on M× [0,T ′].(iv) If T <∞ and limt→TsupM|Rm(x, t)|<∞, then g(t) extends to a smooth solution to9(1.0.1) on M× [0,T1) for some T1 > T so that (ii) is still true with T replacedby T1.From now on, given a bounded curvature complete g0, we will refer to a solu-tion g(t) to (1.0.1) satisfying (i), (ii) as Shi’s solution to (1.0.1). In fact, Chen-Zhu[13] showed that Shi’s solution is unique. More precisely,Theorem 2.1.3 (Chen-Zhu [13]). Let (M,g0) be a complete non-compact Ka¨hlermanifold of bounded curvature, if g1(t), g2(t) are solutions to Ka¨hler-Ricci flow onM× [0,T ] such that gi(t) (i= 1,2) are complete, have uniformly bounded curvatureand gi(0) = g0, then g1(t) = g2(t).Remark 2.1.1. Chen-Zhu’s uniqueness theorem was originally proved for generalreal Ricci flow, the version we stated here is the restriction to the Ka¨hler case.Lott-Zhang [23] also found a similar existence time characterization for Shi’ssolution like Theorem 2.1.1, although it is no longer described by the cohomologyclass of M:Theorem 2.1.4 (Lott-Zhang [23]). Let g0 be a complete Ka¨hler metric with boundedcurvature on a non-compact manifold M. Then the maximal existence time Tg0 of abounded curvature solution is equal to the supremum of the numbers T for whichthere is a bounded function FT ∈C∞(M) such that(i) g0−T Ric(g0)+√−1∂ ∂¯FT ≥ cT g0 for some cT > 0(ii) |FT | and the quantities |∇lRm(g0)|g0 , |∇l∂ ∂¯FT |g0 , for 0 ≤ l ≤ 2, are uni-formly bounded on M.Moreover if T satisfies (i) and (ii), then for any T ′ < T , there is a constant Cdepending T ′, cT , and the bound on the quantities in (ii) such thatC−1g0 ≤ g(t)≤Cg0on M× [0,T ′].In the following sections, we will construct a solution to (1.0.1) for a completeKa¨hler metric g0 that can be approximated by a sequence of complete bounded cur-vature Ka¨hler metrics satisfying some C0 assumptions. Each of sequence members10admits a solution to Ka¨hler-Ricci flow by Shi’s Theorem 2.1.2, then by establishingsome C0 estimates, we can use Evans-Krylov theory to conclude that the sequenceof solutions converges to a limit solution starting from g0. We will also prove auniqueness result of Ka¨hler-Ricci flow which can be applied to unbounded curva-ture metrics satisfying certain conditions.2.2 C0 estimates in terms of background metricsBefore we prove the existence theorems, we need to establish some C0 estimatesfor solutions of Ka¨hler-Ricci flow in terms of a background metric, these estimatesallow us to use the Evans-Krylov theory to conclude that our approximating se-quence converges.In the following, ∇ˆ always denotes the covariant derivative of gˆ.Lemma 2.2.1. Let h(t) be a solution to (1.0.1) on Mn× [0,T0) with h(0) = h0 suchthat h(t) has uniformly bounded curvature on M× [0,T ′] for all 0 < T ′ < T0. Letgˆ be another complete Ka¨hler metric on M with bounded curvature such that theholomorphic bisectional curvature bounded above by K ≥ 0. Let T = 12nK if K > 0,otherwise let T = ∞.(i) Suppose h0 ≥ gˆ. Then h(t)≥(1n −2Kt)gˆ on M× [0,min{T0,T}).(ii) Suppose in addition to (i) we have h0 ≤ Cgˆ, that is, suppose gˆ ≤ h0 ≤ Cgˆ,then(1−w(t))gˆ≤ h(t)≤ (1+w(t))gˆon M× [0,min{T0,T}),where w(t) =√v2(t)(v1(t)+ v2(t)−2n),v1(t) =11n −2Kt,v2(t) = nCe−2κv1(t)tand κ is a lower bound on the bisectional curvature of gˆ. In particular, we havew(0) = n√C(C−1).Proof. (i) Let φ(t) := trh(t)gˆ = h(t)i j¯gˆi j¯. Let 2= ∂∂ t −∆, where ∆ is the Laplacianwith respect to h(t). Then as in [28], we can calculate in a normal coordinate11relative to h(t) and use (1.0.1) to get2φ =((ht)i j¯gˆi j¯)−hkl¯(hi j¯gˆi j¯)kl¯=(Ri j¯gˆi j¯)− (Ri j¯gˆi j¯)+hkl¯hi j¯R̂i j¯kl¯− ĝpq¯hkl¯hi j¯∂kĝiq¯∂l¯ ĝp j¯≤ 2Kφ 2.(2.2.1)Now v1(t) is the positive solution to the ODEdv1(t)dt= 2Kv21(t); v1(0) = nfor t ∈ [0,T ). Let S ∈ (0,min{T0,T}) be fixed. Since h(t) has uniformly boundedcurvature on M× [0,S] we have h(t) ≥C1h0 ≥C1ĝ for some C1 > 0 and hence φis a bounded function on M× [0,S]. Moreover, v1 is also a bounded function onM× [0,S]. Let A = supM×[0,S](φ + v1). Then on M× [0,S]2(e−(2AK+1)t(φ − v1))≤ e−(2AK+1)t [2K(φ 2− v21)− (2AK+1)(φ − v1)]= e−(2AK+1)t [2K(φ + v1)− (2AK+1)] (φ − v1)which is nonpositive at the points where φ−v1≥ 0. Using the fact that h(t) has uni-formly bounded curvature on M× [0,S] and the fact that e−(2AK+1)t(φ − v1)≤ 0 att = 0, which is uniformly bounded on M× [0,S], we conclude that e−(2AK+1)t(φ −v1) ≤ 0 and thus (φ − v1) ≤ 0 on M× [0,S] by the maximum principle, see [24,Theorem 1.2] for example. This proves (i).(ii) Let ψ(t) := trgˆh(t). For any fixed S ∈ [0,min{Th,T}), as in [4] we calculatein a normal coordinate relative to gˆ and use (1.0.1) to get that on M× [0,S):122ψ =(gˆi j¯(ht)i j¯)−hkl¯(gˆi j¯hi j¯)kl¯=− (gˆi j¯Ri j¯)−hkl¯(Rˆi j¯kl¯hi j¯)+(gˆi j¯Ri j¯)− ĝi j¯hpq¯hkl¯∂ihpl¯∂ j¯hkq¯=−hkl¯hi j¯R̂ j¯ikl¯ − ĝi j¯hpq¯hkl¯∂ihpl¯∂ j¯hkq¯≤−2κv1(t)ψ≤−2κv1(S)ψ(2.2.2)by (i). Let wS(t) = nCe−2cv1(S)t be the solution to the ODEdwS(t)dt=−2cv1(S)wS(t); wS(0) = nC.Then arguing as above, we have ψ ≤ wS on Mn × [0,S]. In particular, we getψ(S)≤ wS(S) for every S ∈ [0,min{T0,T}).So far, we have φ(t) ≤ v1(t), and ψ(t) ≤ v2(t) on M× [0,min{T0,T}) wherev1,v2 are as in the statement of the Lemma. Now we follow an idea from [26].At any point in (p, t) ∈ M× [0,min{T0,T}), let λ ′i s be the eigenvalues of h withrespect to gˆ, and calculate at (p, t)n∑i=11λi(1−λi)2 =n∑i=11λi+λi−2=φ +ψ−2n≤v1(t)+ v2(t)−2n(2.2.3)and thus for any fixed i we have−w(t)≤ λi−1≤ w(t) (2.2.4)where w(t) =√v2(t)(v1(t)+ v2(t)−2n). The conclusion in (ii) then follows.The following lemma basically says that if a local solution h(t) to (1.0.1) isa priori uniformly equivalent to a fixed metric gˆ in space time, and close to gˆ at13time t = 0, then it remains close to gˆ in a uniform space time region. Note that incontrast to Lemma 2.2.1, the a priori assumption here is on h(t) for all t.Lemma 2.2.2. Let h(t) be a smooth solution to (1.0.1) on B(1)× [0,T ) with h(0) =h0 where B(1) is the unit Euclidean ball in Cn. Let gˆ be a smooth Ka¨hler metric onB(1). SupposeN−1gˆ≤ h(t)≤ Ngˆ (2.2.5)on B(1)× [0,T ) for some N > 0, and thatgˆ≤ h0 ≤Cgˆ (2.2.6)on B(1). Then there exists a positive continuous function a(t) : [0,T )→R depend-ing only on gˆ,N,C and n such that(1−a(t))Ch0 ≤ h≤ (1+a(t))h0 (2.2.7)on B(1/2)× [0,T ), where a(0) = n√C(C−1).Proof. As in the previous Lemma, let φ = trhgˆ, ψ = trgˆh on B(1)× [0,T0). Choosesome smooth non-negative cutoff function on η : B(1)→ R satisfying η |B(1/2) =1, η |(B(3/4))c = 0, |∇ˆη |2 ≤ C1η , |∂ ∂¯η |gˆ ≤ C2 on B(1) for some constants C1,C2depending only on gˆ. Using the fact that h(t)≥ N−1gˆ, we have|∇η |2 = hi j¯ηiη j¯ ≤ N|∇ˆη |2 ≤ NC1,and|∆η |=∣∣∣hi j¯ηi j¯∣∣∣≤ N|∂ ∂¯η |gˆ ≤ NC2.Now we consider the function ηφ on B(1)× [0,T ). Then in B(1)× [0,T ) at14the point where η > 0, as in the proof of Lemma 2.2.1 (i) we obtain(∂t −∆)(ηφ) = η(∂t −∆)φ −2 < ∇η ,∇φ >−φ∆η= η(∂t −∆)φ −2< ∇η ,∇(ηφ)>η +2|∇η |2ηφ −φ∆η≤ ηC3φ 2−2< ∇η ,∇(ηφ)>η +2NC1φ +NC2φ≤C4−2< ∇η ,∇(ηφ)>η(2.2.8)where the constants C3,C4 depend only on gˆ,N,C and n, where we have used theassumption (2.2.7). Since ηφ is zero outside B(3/4), applying the maximum prin-ciple to ηφ −C4t one can conclude thatηφ ≤ n+C4t =: v˜1(t)on B(1)× [0,T ).Now consider the function ηψ on B(1)× [0,T ). Using the proof of Lemma2.2.1 (ii) and estimating as above we obtainηψ ≤ nC+C5t =: v˜2(t) (2.2.9)on B(1)× [0,T ) for som constants C5 depending only on gˆ,N,C and n.Now at any point in (p, t) ∈ B(1/2)× [0,T ), let λ ′i s be the eigenvalues of hwith respect to gˆ. Then as in the proof of Lemma 2.2.1 (ii) we get that at (p, t)− w˜(t)≤ λi−1≤ w˜(t) (2.2.10)where w˜(t) =√v˜2(t)(v˜1(t)+ v˜2(t)−2n). Since v˜1(0) = n and v˜2(0) = nC, thelemma follows easily from this.In contrast to the previous lemma, in the following lemmas we only assume alower bound on a solution h(x, t) to (1.0.1).Lemma 2.2.3. Let h(x, t) be a smooth solution to (1.0.1) on M× [0,T ) with h(0) =h0. Let p ∈M. Suppose there is a positive continuous function α(t) : [0,T )→ R15such thath(t)≥ α(t)gˆ.where gˆ is a complete Ka¨hler metric with bounded curvature. Then, there exists apositive continuous function β (r, t) : [1,∞)× [0,T )→ R depending only on gˆ theupper bound of trgˆh0 in Bgˆ(p,2r), the lower bound of scalar curvature R(0) of h(0)in Bgˆ(p,2r), α(t) and the dimension n such that for r ≥ 1h(t)≤ β (r, t)gˆ.in Bgˆ(p,r)× [0,T ).Proof. Let d(x) be the distance with respect to gˆ from x to a fixed point p ∈ M.Since gˆ has bounded curvature, by [27] there exists a smooth positive functionρ(x) satisfying d(x)+1≤ ρ(x)≤ d(x)+C on M for some C > 0, with |∇ˆρ|, |∇ˆ2ρ|are bounded on M. Hence without loss of generality, we may assume for simplicitythat d(x) is in fact smooth with |∇ˆd|, |∇ˆ2d| bounded on M.Let φ(s) be smooth function on R such that φ = 1 for s ≤ 1 and is zero fors≥ 2. Moreover, we assume φ ′ ≤ 0, (φ ′)2/φ ≤C1, |φ ′′| ≤C2. Let R be the scalarcurvature of h(t). Then (∂∂ t−∆)R≥ 1nR2. (2.2.11)on M× [0,T ). Let ϕ(x) = φ(d(x)/r). Then ϕ(x) = 0 if d(x) ≥ 2r. Fix someT ′ < T . Then as in the proof of the previous lemma, we compute|∇ϕ|2 = 1r2(φ ′)2|∇d|2=1r2(φ ′)2hi j¯did j¯≤ 1r2α(t)(φ ′)2gˆi j¯did j¯≤C3r2(φ ′)216on B(2r)× [0,T ′] for some constant C3 depending only on T ′,α(t) and gˆ. Similarly,|∆ϕ|=|1rφ ′∆d+1r2φ ′′|∇d|2|≤C4(1r +1r2)on B(2r)× [0,T ′] where C4 depends on C1,C2,T ′,α(t) and gˆ.Now (∂∂ t−∆)(ϕR) =ϕ(∂∂ t−∆)R−R∆ϕ−2〈∇R,∇ϕ〉≥1nϕR2−C5|R|−2〈∇R,∇ϕ〉(2.2.12)on B(2r)× [0,T ′] where C5 depends only on C4 and r. Suppose the infimum of ϕRon B(2r)× [0,T ′] is attained at t = 0, then R ≥ min{0, infBgˆ(p,2r)R(h0)} on Bgˆ(r).Suppose instead that ϕR attains a negative minimum at some (x, t)∈ B(2r)× [0,T ′]where t > 0. Then at (x, t), ∇R =−R∇ϕφ . Hence at this point,0≥1nϕR2−C6|R| (2.2.13)where C6 depends only on C6,C3 and r. Henceϕ2|R| ≤ nC6.on B(2r)× [0,T ′] and we conclude that R≥−C7 on Bgˆ(p,r)× [0,T ′] for some C7depending only on T ′, gˆ,r,α(t). On the other hand,∂∂ tlog(det(hαβ¯ )(t)det(hαβ¯ (0)))=−R≤C7.Sodet(hαβ¯ )(t)det(gˆαβ¯ )≤ eC7t det(hαβ¯ )(0)det(gˆαβ¯ ).on Bgˆ(p,r)× [0,T ′]. Let λi be eigenvalues of h(t)with respect to gˆ. By assumption,λi(x,T ′)≥ α(T ′) for each i and x ∈ Bgˆ(p,r), and the above inequality then implies17λi(x,T ′)≤ β (r,T ′) for some β (r,T ′) depending only on the those constants listedin the Lemma. Moreover, it is not hard to see that β (r,T ′) can be chosen to dependcontinuously on r,T ′ as α(t) is continuous. The Lemma follows as T ′ was chosenarbitrarily.Remark 2.2.1. The completeness assumption of gˆ is only used to ensure the exis-tence of an exhaustion function ρ with bounded gradient and Hessian, it could bedropped if such ρ could be constructed independently.Remark 2.2.2. Given only a local solution h(t) to (1.0.1) on B(1)× [0,T ) whereB(1) is the unit ball on Cn, it is not hard to see from its proof that the conclusionof Lemma 2.2.3 will hold in B(r)× [0,T ) for all r ≤ 1/2.2.3 General existence theoremsWe will now prove the main general existence Theorems for (1.0.1) using the esti-mates in the previous section. Theorems 2.3.1 and 2.3.2 provide general existenceTheorems for (1.0.1) when the initial Ka¨hler metric is realized as a limit of a se-quence of Ka¨hler metrics satisfying certain properties. In fact, our initial metricmay have unbounded curvature or may even be only Hermitian continuous withcurvature undefined. When the initial metric is only Hermitian, the convergence ofthe solution g(t) at time zero has to be understood in the C0 sense.In the following, we say that a sequence of smooth metrics hk converge uni-formly (resp. smoothly) to a metric g on a set U , if hk converge to g in the C0 (resp.Ck for all k) norm on U .Theorem 2.3.1. Let g0 be a complete continuous Hermitian metric on a noncom-pact complex manifold Mn. Suppose there exists a sequence {hk,0} of smoothcomplete Ka¨hler metrics with bounded curvature on M converging uniformly oncompact subsets to g0 and another complete Ka¨hler metric ĝ on M with boundedcurvature and holomorphic bisectional curvature bounded from above by K ≥ 0such that(i) hk,0 ≥ gˆ for all k;18(ii) for every k, the Ka¨hler-Ricci flow (1.0.1) has smooth solution hk(t) with ini-tial data hk,0 on M× [0,T ′) for some T ′ > 0 independent of k such that thecurvature of hk(t) is uniformly bounded on M× [0,T1] for all 0 < T1 < T ′;(iii) The scalar curvature Rk of hk,0 satisfies: for any r > 0, there exists a constantCr > 0 such that Rk ≥−Cr on Bgˆ(p,r) for some fixed point p ∈M and all k.Let T = min{T ′, 12nK} if K > 0, otherwise let T = T ′. Then the Ka¨hler-Ricci flow(1.0.1) has a complete smooth solution g(t) on M× (0,T ) which extends continu-ously to M× [0,T ) with g(0) = g0 and satisfies g(t)≥ (1/n−2nK)gˆ on M×(0,T ).Moreover, if g0 is smooth and {hk,0} converges smoothly and uniformly oncompact subsets of M, then g(t) extends to a smooth solution to (1.0.1) on M×[0,T ) with g(0) = g0.Proof. By Lemma 2.2.1, we havehk(t)≥(1n−2Kt)gˆ (2.3.1)as long as t < T0 = 1/(2nK). By Theorem 2.1.2, let gˆ(t) be the solution Ka¨hler-Ricci flow in the theorem with initial condition gˆ. Then for any 1 > ε > 0 small,choose 0 < t0 small enough so that (1− ε)ĝ(t0)≤ ĝ≤ (1+ ε)ĝ(t0). Then we havehk(t)≥(1n−2Kt)(1− ε)gˆ(t0) (2.3.2)and gˆ(t0) has bounded geometry of infinite order. By Lemma 2.2.3, for there is apositive continuous function β (r, t) : [1,∞)× [0,T0)→ R such that for r ≥ 1hk(t)≤ β (r, t)gˆ(t0). (2.3.3)in Bˆ(p,r)× [0,T )where T =min{T ′, 12nK} and p∈M is a fixed point. We concludefrom Theorem A.1.1 (i), that passing to some subsequence, the hk(t)’s converge toa solution g(t) of Ka¨hler-Ricci on M× (0,T ) so that (2.3.1) is true. Moreover, ifg0 is smooth and {hk} converges smoothly and uniformly to g0 on compact sets,then we see from Theorem A.1.1 (ii) that in fact g(t) extends to a smooth solutionon M× [0,T ) such that g(0) = g0.19We now prove g(t) converge uniformly on compact set to g0 as t → 0 wheng0 is only assumed to be continuous. Fix any x ∈M and a local biholomorphismφ : B(1)→ M where B(1) is the open unit ball in Cn, and φ(0) = x. Considerthe pullbacks φ ∗hk(t), φ ∗hk = φ ∗hk(0), φ ∗gˆ, which by abuse of notation we willsimply denote by hk(t), hk, gˆ, respectively, for the remainder of proof. In particular,hk(t) solves Ka¨hler-Ricci flow (1.0.1) on B(1)× [0,T ).Now by our hypothesis on the convergence of hk, given any δ > 0 we may findk0 such that |hk0,0−g0|gˆ ≤ δ and(1−δ )hk0,0 ≤ hk,0 ≤ (1+δ )hk0,0 (2.3.4)for all k≥ k0. On the other hand, by (2.3.2) and (2.3.3) we can find N > 0 such thatN−1hk0,0 ≤ hk(t)≤ Nhk0,0 (2.3.5)in B(1)× [0,T/2) for all k ≥ k0. Then by Lemma 2.2.2, there exists a continuousfunction a(t) depending on N,hk0 and δ such that(1−a(t))(1−δ )2(1+δ )hk0,0 ≤ hk(t)≤ (1+a(t))(1+δ )hk0,0in B(12)× [0,T/2) with a(0) = n√C(C−1), with C = (1+δ )/(1−δ ). Note thata(t) is independent of k. Letting k→ ∞ gives(1−a(t))(1−δ )2(1+δ )hk0,0 ≤ g(t)≤ (1+a(t))(1+δ )hk0,0 (2.3.6)in B(12)× (0,T/2). We then getlimsupt→0|g(t)−g0|gˆ≤ limsupt→0(|g(t)−hk0,0|gˆ+ |hk0,0−g0|gˆ)≤[∣∣∣∣1− (1−a(0))(1−δ )2(1+δ )∣∣∣∣+ |(1+a(0))(1+δ )−1|] |hk0,0|ĝ+δ |hk0,0|ĝ20uniformly on B(12). Then letting δ → 0 above, and using the fact that a(0)→ 0 asδ → 0, and (2.3.2) and (2.3.3) we conclude thatlimsupt→0|g(t)−g0|gˆ = 0.uniformly on B(12). Hence g(t) converge to g0 uniformly on compact sets as t→ 0.We do not have any bound on the curvature of the solution g(t) in the previ-ous theorem. Also in the previous theorem, we assume that the Ka¨hler-Ricci flow(1.0.1) has solution with initial condition hk,0 on a fixed time interval independentof k. We want to remove this assumption and obtain curvature bound for the solu-tions. In order to do this, we assume hk,0 also has an uniform upper bound.Theorem 2.3.2. Let g0 be a complete continuous Hermitian metric on a noncom-pact complex manifold Mn. Suppose there exists a sequence {hk,0} of smoothcomplete Ka¨hler metrics with bounded curvature on M converging uniformly oncompact subsets to g0 and another complete Ka¨hler metric ĝ on M with boundedcurvature and holomorphic sectional curvature bounded from above by K ≥ 0 suchthat(i) C−1gˆ≤ hk,0 ≤Cgˆ for some C independent of k;(ii) hk has bounded curvature for every k.Let T = 1/(2CnK) if K > 0, otherwise let T = ∞. Then the Ka¨hler-Ricci flow(1.0.1) has a smooth solution g(t) on M× (0,T ) such that(a) (1/(nC)−2Kt)gˆ≤ g(t)≤ B(t)gˆ on M× (0,T ) for some positive continuousfunction B(t) depending only on C, gˆ and n.(b) g(t) has bounded curvature for t > 0. More precisely, for any 0 < T ′ < Tand for any l ≥ 0 there exists a constant Cl depending only on C, l, T ′, gˆ andthe dimension n such thatsupM|∇lRm(g(t))|2g(t) ≤Clt l+2,21(c) g(t) converges uniformly on compact subsets to g0 as t→ 0.Moreover, if g0 is smooth and {hk,0} converges smoothly and uniformly oncompact subsets of M, then g(t) extends to a smooth solution on M× [0,T ) withg(0) = g0.Proof. For each k, let hk(t) be the solution to (1.0.1) with initial condition hk fromTheorem 2.1.2 which is defined on M× [0,Tk) for some Tk > 0. We first claimthat there is T > 0 such that Tk ≥ T for all k. By Lemma 2.2.1, there is a positivecontinuous function B(t) : [0,T )→ R independent of k such that(1/n−2CKt)gˆ≤ hk(t)≤ B(t)gˆin M× [0,min{Tk,T}) where T = 1/(2nCK). As before, we may assume that gˆhas bounded geometry of infinite order. By Theorem A.1.1, we conclude that ifTk < T , then |Rm(hk(t))|hk(t) are bounded in M× [0,Tk). By Theorem 2.1.2, wesee that one can extend hk(t) so that Tk ≥ T for all k as claimed. Given upper andlower bounds on hk(t) as above, we may conclude from Theorem A.1.1, as in theproof of Theorem 2.3.1, that there is a smooth solution to the Ka¨hler-Ricci flowg(t) on M× (0,T ) satisfying condition (a) and (c) from which we conclude, byTheorem A.1.1 (i), that condition (b) is also satisfied.2.4 A uniqueness theorem and an existence time estimateIn this section we will discuss some results on existence time and uniqueness ofthe complete bounded curvature solutions and uniqueness of solutions in general.We first prove a uniqueness theorem which could be applied to solutions that maynot have bounded curvature:Theorem 2.4.1. Let (Mn, ĝ) be a complete non-compact Ka¨hler manifold. Supposethere is an exhaustion function ζ > 0 on (Mn, ĝ) with limx→∞ ζ (x) = ∞ such that|∂ ∂¯ζ |ĝ and |∇̂ζ |ĝ are bounded.Let g1(x, t) and g2(x, t) be two solutions of the Ka¨hler-Ricci flow (1.0.1) onM× [0,T ] with the same initial data g0(x) = g1(x,0) = g2(x,0). Suppose there is22a positive function σ with limx→∞ logσ(x)/ logζ (x) = 0 such that the followingconditions hold for all (x, t) ∈M× [0,T ]:(i)ĝ(x)≤ ζ (x)g1(x, t); ĝ(x)≤ ζ (x)g2(x, t), (2.4.1)(ii)−σ(x)≤ det((g1)i j¯(x, t))det((g2)i j¯(x, t))≤ σ(x).Then g1 ≡ g2 on M× [0,T ]. In particular, if g1 and g2 are uniformly equivalent toĝ on M× [0,T ], then g1 ≡ g2.Proof. By adding a positive constant to ζ we may assume that η := logζ > 1.Thenηi j¯ =ζi j¯ζ− ζiζ j¯ζ 2.Since |∂ ∂¯η |ĝ and |∇̂η |ĝ are uniformly bounded, there is c1 > 0 such that|∂ ∂¯η |ĝ ≤ c1ζon M. Let h(s, t)(x) = sg1(x, t) + (1− s)g2(x, t), 0 ≤ s ≤ 1. By (i), we haveĝ(x) ≤ ζ (x)h(s, t)(x) for all (x, t) ∈ M× [0,T ] and for all s. Let (x, t) be fixedand diagonalize ∂ ∂¯η with respect to ĝ at x. Then |ηii¯| ≤ c1ζ . On the other hand,∆h(s,t)η = (h(s, t))i j¯ηi j¯ = (h(s, t))ii¯ηii¯ ≤ nζ ·c1ζ= nc1. (2.4.2)Letw(x, t) =∫ t0(logdet((g1)i j¯(x,s))det((g2)i j¯(x,s)))ds.Thenwi j¯(x, t) =∫ t0((R1)i j¯(x,s)− (R2)i j¯(x,s))ds =−(g1)i j¯(x, t)+(g2)i j¯(x, t)where (Rk)i j¯ is the Ricci tensor of gk, k = 1,2. Here we have used the Ka¨hler-Ricciflow and the fact that g1 = g2 at t = 0. Hence in order to prove the proposition, it23is sufficient to prove that w≡ 0. Now∂∂ tw(x, t) =∫ 10∂∂ slogdet(hi j¯(s, t)(x)))ds=∫ 10∆h(t,s)w(x, t)ds.(2.4.3)Let W (x, t) = eAtη where A = nc1+1. By (2.4.2),∂∂ tW (x, t)−∫ 10∆h(t,s)W (x, t)ds≥eAt(Aη−nc1)≥eAtηwhere we have used the fact that η > 1. For any ε > 0,∂∂ t(εW −w)(x, t)−∫ 10∆h(t,s) (εW −w)(x, t)ds≥eAtηBy (ii), limx→∞(εW −w)(x, t) = ∞ uniformly in t. By the maximum principle,we conclude that w ≤ εW . Letting ε → 0 gives w ≤ 0. Similarly, one can provethat −w ≤ 0 and hence w ≡ 0 on M× [0,T ]. This completes the proof of theproposition.As a corollary, we haveCorollary 2.4.1. Let (Mn, gˆ) be a complete non-compact Ka¨hler manifold withbounded curvature. Let g1(x, t) and g2(x, t) be two solutions of the Ka¨hler-Ricciflow (1.0.1) on M× [0,T ] with the same initial data g0(x) = g1(x,0) = g2(x,0).Suppose there is a constant C such that:C−1gˆ≤ g1(t),g2(t)≤Cgˆon M× [0,T ]. Then g1 ≡ g2 on M× [0,T ].By using Theorem 2.1.4, we can compare the maximal existence time of theKa¨hler-Ricci flow solutons by comparing the C0 data of the initial metric. Forconvenience, if g is a complete Ka¨hler metric with bounded curvature, we let Tg bethe maximal existence time of Shi’s solution.24Theorem 2.4.2. Let M be a non-compact complex manifold, g0 and h0 be completeKa¨hler metrics with bounded curvature. If g0 and h0 are equivalent and g0 ≥ ah0then Tg0 ≥ aTh0 ,Proof. First observe that if λ > 0 is a constant, then Tλh0 = λTh0 . Hence withoutloss of generality, we may assume that a = 1. Also, we assume without loss ofgenerality that g0,h0 have bounded geometry of order infinity. For if not, we letg(t),h(t) are the corresponding solutions as in Theorem 2.1.2, then we first provethe Theorem for g(ε),h(ε) for arbitrary small ε and then let ε→ 0 and use the factthat g(ε),h(ε) converge uniformly to g0,h0 (respectively) on M, and Tg(ε) = Tg−εand Th(ε) = Th−ε by Theorem 2.4.1. Note that by Theorem A.1.1, all the covariantderivatives of g(ε) with respect to h0 are bounded. So we may assume in additionthat all the covariant derivatives of g0 with respect to h0 are bounded.Now for any 0 < T < Th0 , we haveg0−T Ric(g0) = h0−T Ric(h0)+T (Ric(g0)−Ric(h0))+(g0−h0).By Theorem 2.1.4, there is a smooth bounded function f with bounded covariantderivatives with respect to h0 such thath0−T Ric(h0)+√−1∂ ∂¯ f ≥C1h0for some C1 > 0. Then letting F = logωn0ηn where ω0 and η are the Ka¨hler forms ofg0 and h0 respectively, givesg0−T Ric(g0)+√−1∂ ∂¯ ( f +T F)≥C1h0+(g0−h0)≥C2g0for some C2 > 0 because g0 ≥ h0 and g0 is uniformly equivalent to h0. Fromthe facts that g0,h0 are equivalent and that all the covariant derivatives of g0 withrespect to h0 are bounded, we may conclude that all the covariant derivatives of fare bounded with respect to g0 as well. We may also conclude from these facts thatF and |∇l∂ ∂¯F |g0 are uniformly bounded for 0 ≤ l ≤ 2 where we have used that√−1∂ ∂¯F = −Ric(g0)+Ric(h0). By Theorem 2.1.4, we conclude that T ≤ Tg0 .25From this the result follows.By the theorem, we have the following monotonicity and continuity of Tg.NamelyCorollary 2.4.2. Let Mn be a non-compact complex manifold.(i) Let g0 ≥ h0 be complete uniformly equivalent Ka¨hler metrics on M withbounded curvature. Then Tg0 ≥ Th0 . In particular, if Th0 = ∞, then Tg0 = ∞.(ii) LetK be the set of complete Ka¨hler metrics on M with bounded curvature.Then Tg is continuous on K with respect to the C0 norm in the followingsense: Let g0 ∈K . Then for h0 ∈K , Th0 → Tg0 as ||h0−g0||g0 → 0.As a corollary, instead of the C0 data, we can also estimate the existence by theupper bound of the Ricci curvature of another equivalent Ka¨hler metric.Corollary 2.4.3. Let (M,h) be a complete Ka¨hler manifold having bounded cur-vature with Ricci curvature bounded above by K. If g is equivalent to h and g≥ h.then Tg ≥ 1K .Proof. Since Ric(h)≤Kh, so h− tRic(h) = (1−Kt)h. By Theorem 2.1.4, Th ≥ 1K .By Theorem 2.4.2, Tg ≥ 1K .Remark 2.4.1. As an example of application, let (M,h) be a complete Ka¨hler man-ifold with non-positive bisectional curvature (e.g. an Euclidean space or a Poincare´disk), assuming that g is equivalent to h, since K = 0 we have Tg = ∞.26Chapter 3U(n)-invariant Ka¨hler metricsIn this chapter we will study the Ka¨hler-Ricci flow starting from U(n)-invariantKa¨hler metrics, the main theorem isTheorem 3.0.1. Let g0 be a complete U(n)-invariant Ka¨hler metric on Cn withnon-negative bisectional curvature. Then(i) the Ka¨hler-Ricci flow (1.0.1) has a unique smooth longtime U(n)-invariantsolution g(t) which is equivalent to g0 and has bounded non-negative bisec-tional curvature for all t > 0;(ii) g(t) converges, after rescaling at the origin, to the standard Euclidean metricon Cn.Here we say that the solution g(t) converges to g˜ after rescaling at a point p iffor some V ∈ TpM, the metrics 1|V |2t g(t) converges to g˜ smoothly and uniformly oncompact subsets on M where |V |2t = gt(V,V ).The content is organized as follows: We first collect some results about U(n)-invariant Ka¨hler metrics. These results correlate U(n)-invariant metrics with non-negative holomorphic bisectional curvature with non-decreasing real valued func-tions. Then we will construct a short-time U(n)-invariant Ka¨hler-Ricci flow solu-tion starting from any non-negatively curved U(n)-invariant metric. After that, wewill show that the short-time solution can be extended to a long-time solution and27the solution is unique in a certain class. Finally we will show that the long-timesolution converges to the standard Euclidean metric after rescaling.3.1 Background materialsLet g be a U(n)-invariant Ka¨hler metric. Then there is a smooth p ∈ C∞[0,∞)such that the Ka¨hler form of the metric satisfies ω = i∂ ∂¯ p(r) where r = |z|2. Inthe standard coordinate on Cn, gi j¯ = f (r)δi j + f ′(r)z¯iz j where f (r) = p′(r). Leth = (r f )′, then at the point (z1,0, · · · ,0),g11¯ = h,gii¯ = f and gi j¯ = 0 for all i 6= j.Due to U(n)-invariance, this basically describes the metric on the whole of Cn.Using this observation, Wu-Zheng [33] parametrized the U(n)-invariant Ka¨hlermetrics by a constant C and a smooth function ξ : [0,∞)→ R as follows:Theorem 3.1.1. (a) ([WU-ZHENG] [33]) Every smooth U(n) invariant Ka¨hlermetric g is generated by a function ξ : [0,∞)→ R with ξ (0) = 0 such that ifhξ (r) :=Ce∫ r0 − ξ (s)s ds; fξ (r) :=1r∫ r0hξ (s)dswhere hξ (0) =C > 0 and fξ (0) = hξ (0), where r = |z|2, thengi j¯ = fξ (r)δi j + f′ξ (r)ziz j.where gi j¯ are the components of g in the standard coordinates z= (z1, . . . ,zn)on Cn. Moreover g is complete if and only if∫ ∞0√hξ (s)√sds = ∞.(b) ([WU-ZHENG] [33]) Let h = hξ , f = fξ . At the point z = (z1,0, . . . ,0),relative to the orthonormal frame e1 = 1√h∂z1 ,ei =1√f ∂zi , i≥ 2, with respectto g, the curvature tensor has componentsA = R11¯11¯ =ξ ′h,28B = R11¯ii¯ =1(r f (r))2∫ r0ξ ′(s)(∫ t0h(s)ds)dt,C = Rii¯ii¯ = 2Rii¯ j j¯ =2(r f (r))2∫ r0h(s)ξ (s)dt,where 2 ≤ i 6= j ≤ n and these are the only non-zero components of thecurvature tensor at z except those obtained from A,B or C by the symmetricproperties of the curvature tensor.(c) ([WU-ZHENG] [33], YANG [34]) g has positive (nonnegative) bisectionalcurvature if and only if ξ ′ > 0 (ξ ′ ≥ 0). In particular, if g has nonnegativebisectional curvature and is complete, then ξ ≤ 1.Remark 3.1.1. If g1 and g2 are two smooth U(n) invariant Ka¨hler metrics on Cngenerated by ξ1,ξ2 respectively, and if the corresponding functions hξ1 , hξ2 satisfyhξ1 ≥ hξ2 , then g1 ≥ g2. Conversely, if g1 ≥ g2, then hξ1 ≥ hξ2 . This can be seen bycomparing the metrics at the points (a,0, . . . ,0).Using Theorem 3.1.1 (a) and (b), we can find a condition for which the curva-ture of g is uniformly bounded.Lemma 3.1.1. Let g be a complete U(n) invariant Ka¨hler metric on Cn generatedby ξ . If∣∣∣ ξ ′h ∣∣∣ is uniformly bounded, then the curvature of g is uniformly bounded.Proof. It is sufficient to prove that the holomorphic bisectional curvature is uni-formly bounded under the assumption that∣∣∣ ξ ′h ∣∣∣ is uniformly bounded by c, say.By Theorem 3.1.1, in the notations of the theorem it is sufficient to prove that|A|, |B|, |C| are uniformly bounded. It is obviously |A| ≤ c. Now|B| ≤ 1r2 f 2∫ r0ch(t)dt(∫ t0h(s)ds)dt≤ cr2 f 2(∫ r0h(t)dt)2=cbecause h > 0 and r f (r) =∫ r0 h(t)dt. Similarly, since|ξ (r)| ≤∫ r0|ξ ′(t)|dt ≤ c∫ r0h(t)dt,29we have|C| ≤ 2c.Theorem 3.1.1 (c) showed that any complete U(n)-invariant Ka¨hler metric withnon-negative bisectional curvature corresponds to a non-decreasing ξ with ξ (∞) =limr→∞ ξ (r) ≤ 1. In fact, the volume growth rate can also be described by ξ .Chen-Zhu [13] proved that if (M,g) is complete non-compact with non-negativebisectional curvature, then for all x0 ∈M, there exists c > 0 such that1csn ≤V (s)≤ cs2n, ∀s > 1,where V (s) be the volume of Bg(x0,s).This means that complete non-negatively curved Ka¨hler manifolds have vol-ume growth rate between half-Euclidean and Euclidean. In case the volume growthrate is half-Euclidean (i.e. V (s) is asymptotic to sn), the space is called a cigar; ifthe volume growth rate is Euclidean (i.e. V (s) is asymptotic to s2n), the space iscalled a conoid. In the U(n)-invariant case, conoids and cigars satisfy the followingconditions:Theorem 3.1.2 (Wu-Zheng, [33]). The metric ω is a conoid if the correspondingξ satisfies ξ (∞)< 1. It is a cigar if∫ ∞11−ξr dr < ∞.Using the construction of Cabezas-Rivas and Wilking [2], Yang-Zheng [35]proved the short time existence of the Ka¨hler-Ricci flow for complete non-collapsedU(n)-invariant metric with non-negative sectional curvature. Their solution isU(n)-invariant when some technical assumptions on initial data is satisfied, andwhen the solution is U(n) invariant, they proved the following theorem which wewill use later:Theorem 3.1.3 (Yang-Zheng, [35]). Let g(t), t ∈ [0,T ] be a complete solution ofthe Ka¨hler-Ricci flow on Cn with U(n)-symmetry. If g(0) has non-negative holo-morphic bisectional curvature, so does g(t) for all t ∈ [0,T ].In the following content, by rescaling the initial metric if necessary, we willassume that the constant C of g0 in Theorem 3.1.1(a) is equal to 1.303.2 Short time existence of U(n)-invariant Ka¨hler-RicciflowWe will use Theorem 2.3.2 to prove short time existence. We first prove a Proposi-tion:Proposition 3.2.1. Assume that g0 is a complete U(n)-invariant metric with non-negative bisectional curvature ξ , there exists a complete U(n) invariant metric gˆwith bounded curvature andc−1gˆ≤ g0 ≤ cgˆ (3.2.1)on Cn for some constant c > 0.Proof. We will prove the Theorem by constructing a ξˆ satisfying certain proper-ties, the constant C of ĝ in Theorem 3.1.1(a) is taken to be 1. Let ξ and ξˆ corre-spond to g0 and gˆ respectively, if ξ ≡ 0, then g0 is Euclidean and so the statementis true by taking ĝ = g0.Suppose now ξ 6= 0, we consider two cases:Case 1: If∫ r1ξ−1t dt ≥ −C for some constant C. Let ξˆ be a smooth functionon [0,∞) such that ξˆ (r) = 1 for all sufficiently large r, it is clear that it generates acomplete bounded curvature ĝ. By the definition of h, for all r ≥ 0, we havehˆ(r)h(r)= exp(∫ r0ξ (t)− ξˆ (t)tdt).Because∫ r1ξ−1t dt ≥ −C, there exists C′ such that∫ r0ξ (t)−ξˆ (t)t dt ≥ −C′ and there-fore h(r) ≤ C2hˆ(r) for some constant C2. On the other hand, because ξ ≤ 1 andξˆ (r) = 1 for all sufficiently large r, therefore, h(r) ≥ C1hˆ(r) for some C1. SinceC1hˆ(r) ≤ h(r) ≤ C2hˆ(r) for all r, by Remark 3.1.1, we have C1gˆ ≤ g0 ≤ C2gˆ asclaimed.Case 2: If∫ r1ξ−1t dt is not bounded from below. By the assumption ξ 6= 0 andξ is non-decreasing,∫ r1ξt dt is not bounded from above. We want to construct ξˆ byoscillating between 0 and 1 at a suitable rate so that the corresponding metric ĝ iscomplete with bounded curvature and is equivalent to g. More precisely, we will31find ξˆ and 1≤ a0 < a1 < a2 · · · → ∞ such that ξˆ generates a complete U(n) metricĝ such that ∫ a2(i+1)a2iξ − ξˆtdt = 0 (3.2.2)for all i≥ 0; ∣∣∣∣∣∫ ra2iξ − ξˆtdt∣∣∣∣∣≤ c1 (3.2.3)for some c1 for all i≥ 0 and for all r ∈ [a2i,a2(i+1)); and∣∣∣∣∣ ξˆ ′(r)hˆ(r)∣∣∣∣∣≤ c2 (3.2.4)for some c2 for all r ≥ 0. Then by Lemma 3.1.1, Theorem 3.1.1 (a), we can con-clude that ĝ satisfies the conditions of the Proposition.Fix a smooth function ρ on R, such thatρ(t) ={1, if t ≤ 1+ ε;0, if t ≥ 3− ε ,and ρ ′ ≤ 0, where ε > 0 is small enough so that 1+ ε < 3− ε . Then 0≤ ρ ≤ 1.Let ξˆ be a smooth function on [0,1] with ξˆ (0) = 0 and ξˆ (r) = 1 near r = 1such that 0≤ ξˆ ≤ 1. We are going to find ai and ξˆ (r) on [ai,ai+1] inductively. Leta0 = 1. ∫ 3a0a0ξ −ρ( ta0 )tdt ≤∫ 3a0a01−ρ( ta0 )tdt ≤ log3.Since∫ r3a0ξt dt is not bounded from above, there is a first a1 > 3a0 such that∫ 3a0a0ξ −ρtdt+∫ a13a0ξtdt = c3where c3 = log3+1. On the other hand,∫ 3a1a1ξ − (1−ρ( ta1 ))tdt ≥− log3.32Since∫ r3a1ξ−1t dt is not bounded from below, there exists a first a2 > 3a1, such that∫ 3a1a1ξ − (1−ρ( ta1 ))tdt+∫ a23a1ξ −1tdt =−c3Defineξˆ (r) =ρ( ra0 ), if a0 ≤ r ≤ 3a0;0, if 3a0 < r ≤ a1;1−ρ( ra1 ), if a1 < r ≤ 3a1;1, if 3a1 < r ≤ a2.It is easy to see that ξˆ is smooth on [0,a2] with ξˆ (r) = 1 near a2. Moreover,0≤ ξˆ ≤ 1 on [1,a2], and ∫ a2a0ξ − ξˆtdt = 0.so (3.2.2) is true for i = 0. It is easy to see that|ξ ′| ≤ c4rwhere c4 = 3max |ρ ′|.For a0 ≤ r ≤ a1, by the definition of a1 we have∫ ra0ξ − ξˆtdt ≤ c3.For a1 < r ≤ a2,∫ ra0ξ − ξˆtdt =(∫ a1a0+∫ ra1)ξ − ξˆtdt≤c3+∫ ra11− ξˆtdt≤c3+ log3.Hence for a0 ≤ r ≤ a2, ∫ ra0ξ − ξˆtdt ≤ 2c3.33Similarly, one can prove that∫ ra0ξ − ξˆtdt ≥−2c3.To summarize, we have find ξˆ (r) and a0 < a1 < a2 such that ξˆ is smooth anddefined on [0,a2] with ≤ ξˆ ≤ 1 on [a0,a2], satisfying (3.2.2) with i = 0, (3.2.3)with i = 0, c1 = 2c3, and |ξˆ ′| ≤ c4r on [a0,a2]. Moreover, ξˆ (r) = 1 near r = a2.From the above construction, it is easy to see that one can continue and finda2 < a3 < a4 · · · → ∞ and ξˆ with 0 ≤ ξˆ (r) ≤ 1 for r ≥ a0, satisfying (3.2.2) and(3.2.3) with c1 = 2c3, and |ξˆ ′| ≤ c4r on [a0,∞).Since ξˆ ≤ 1,hˆ(r)≥ c5 exp(−∫ r11tdt)≥ c5rfor some c5 > 0 for all r ≥ 1. Combing with the fact that |ξˆ ′| ≤ c4r on [a0,∞), weconclude that (3.2.4) is also true. This completes the proof.We also need the following Lemma:Lemma 3.2.1. Let g0 be a complete U(n)-invariant metric with non-negative bi-sectional curvature and let gˆ be a complete bounded curvature U(n)-invariant met-ric equivalent to g0. Then we can find a sequence of complete U(n)-invariant met-ric hk such that1. hk converges to g0 in C∞loc-sense;2. there exists a uniform constant C such that 1C ĝ≤ hk ≤Cĝ and3. hk has bounded curvature for all k,Proof. Choose δk > 0 and smooth functions ηk : (−∞,∞)→ R satisfyingηk(r) := 1 if −∞< r ≤ k0 < ηk(r)< 1 if k < r < k+δk= 0 if k+δk ≤ r < ∞.(3.2.5)34and ∫ k+δkk∣∣∣∣∣(ξ − ξˆ )t∣∣∣∣∣dt ≤ 1 (3.2.6)for all k. Let {ξk} : [0,∞)→ ∞ be defined byξk(r) = ηkξ +(1−ηk)ξˆ .Then each ξk generates a U(n) invariant Ka¨hler metric hk and for all r ≥ 0,∫ r0ξk(t)− ξˆ (t)tdt =∫ r0ηk(ξ − ξˆ )tdt={ ∫ r0ξ−ξˆt dt, if r ≤ k;∫ k0ξ−ξˆt dt+αk, if r > kwhere|αk| ≤∫ k+δkk∣∣∣∣∣ξ − ξˆt∣∣∣∣∣dt ≤ 1.By Remark 3.1.1, since g and ĝ are equivalent, there exists constant C such that|∫ r0 ξ−ξˆt dt| ≤C. Combining the above inequalities and use Remark 3.1.1 again, wehavec−12 gˆ≤ hk ≤ c2gˆ (3.2.7)for some constant c2 > 0, for all k. Since ξk = 1(r) for all sufficiently large r, hk iscomplete with bounded curvature. And because ξk = ξ (and hence hk = g0) on theset {r < k}, we have hk converges to g0 in C∞loc-sense.Now we are ready to prove a short time existence theorem for U(n)-invariantKa¨hler-Ricci flow:Theorem 3.2.1. Let g0 be a complete U(n)-invariant Ka¨hler metric on Cn withnon-negative bisectional curvature. Then for some T > 0 the Ka¨hler-Ricci flow(1.0.1) has a complete smooth U(n)-invariant solution g(t) on Cn× [0,T ) withg(0) = g0. Moreover, for every l ≥ 0 there exists a constant cl depending only on35such thatsupp∈Cn‖∇lRm(p, t)‖2t ≤clt l+2(3.2.8)on Cn× (0,T ).Proof. Let gˆ be the U(n) invariant Ka¨hler metric with bounded curvature generatedby ξˆ defined in Proposition 3.2.1, so thatc−11 gˆ≤ g0 ≤ c1gˆ (3.2.9)for some c1 > 0 as in Proposition 3.2.1. Also let hk be the U(n) invariant Ka¨hlermetric with bounded curvature generated by ξk defined in Lemma 3.2.1. In partic-ular, hk is complete with bounded curvature, converges in C∞loc to g0 andc−12 gˆ≤ hk ≤ c2gˆ (3.2.10)for some constant c2 > 0, for all k. Now recall that the curvature of gˆ is boundedby a constant K as in Proposition 3.2.1, and thus by Theorem 2.1.2 we may assumewithout loss of generality that gˆ has bounded geometry of order infinity. By Theo-rem 2.3.2, there is a solution g(t) of the Ka¨hler-Ricci flow with initial condition g0on M× [0,T ) for some T > 0 so that||Rm(g(t))||2g(t) ≤c3tfor some c3 > 0 and for all 0 < t < T . The estimates for ||∇lRm|| for each l ≥ 0then follows from the general results of [26].Recall that in Theorem 2.3.2, g(t) is obtained as the limit of hk(t), since hk(t)is U(n)-invariant, so g(t) is also U(n)-invariant.Combining Theorem 3.2.1 with Theorem 3.1.3, we haveCorollary 3.2.1. Let g0 be a complete U(n)-invariant metric with non-negative bi-sectional curvature, there exists a complete U(n)-invariant metric g1 with boundednon-negative bisectional curvature such that g1 and g0 are equivalent.Proof. As shown in the proof of Theorem 3.2.1, there is a background metric ĝwhich is complete with bounded curvature and a short time U(n)-invariant solution36g(t) on [0,T ] such that 1c2 ĝ ≤ g(t) ≤ c2ĝ for all t ∈ [0,T ]. It suffices to take g1 tobe g(ε) where 0 < ε ≤ T , the curvature bound follows from Theorem 3.2.1 and thenon-negativity of bisectional curvature is a consequence of Theorem 3.1.3.3.3 Long time solutionThe main theorem of this section is :Theorem 3.3.1. Let g0 be a complete U(n)-invariant Ka¨hler metric on Cn withnon-negative holomorphic bisectional curvature. Then the Ka¨hler-Ricci flow (1.0.1)has a unique long time U(n)-invariant solution g(t) which is equivalent to g0 andhas bounded non-negative bisectional curvature for all t > 0.Recall that g(t) and g0 are equivalent if for all T < ∞, there exists constant Csuch that 1C g0 ≤ g(t)≤Cg0 for all t ∈ [0,T ]. The uniqueness is an immediate con-sequence. In fact, suppose g1(t) and g2(t) satisfy Theorem 3.3.1, by Proposition3.2.1, there is a complete bounded curvature ĝ equivalent to g0, hence ĝ is equiv-alent to both g1(t) and g2(t). The uniqueness then follows from Corollary 2.4.1immediately.In Theorem 3.2.1, we already obtained a short-time solution g(t) satisfying theconditions and by the uniqueness argument above, we only need to extend it to along time solution which is equivalent to g0. And replacing g0 by g(ε) if necessary,we can assume g0 has bounded curvature.We will prove the theorem by considering two cases:Case 1: When g0 is a cigar (i.e. g0 has half-Euclidean volume growth).Proof. Let h be Cao’s cigar constructed in [6]. Recall that it is a complete U(n)-invariant metric which has positive bisectional curvature everywhere and the scalarcurvature decays like 1ρ , where ρ is the distance to the origin with respect to h.It is a soliton so that there is a Ricci flow solution h(t) on [0,∞) such that h(t) isisometric to h(0) for all t ∈ [0,∞). Since h has positive bisectional curvature andits scalar curvature decays linearly, h(t) is a complete bounded curvature long timesolution to Ricci flow. Suppose ξ and ξh corresponds to g0 and h respectively, by37Theorem 3.1.2 and the fact that ξ , ξh are both non-decreasing, we have∣∣∣∣∫ r0 ξ −ξht dt∣∣∣∣≤C0+∫ ∞1 1−ξt dt+∫ ∞11−ξhtdt ≤C1where C0, C1 are constants independent of r. In particular, g0 and h are equivalent,so by Corollary 2.4.2 (i), g0 admits a long-time complete bounded curvature solu-tion g(t). And since g(t) has bounded curvature, it must be equivalent to g0. Tosee that g(t) is U(n)-invariant, we observe that g(t) coincides with the short timeU(n)-invariant solution obtained in Theorem 3.2.1 due to the uniqueness theorem.Let T be the supremum of time such that g(t) remains U(n)-invariant on [0,T ) andsuppose that T < ∞. Since g(t) is a long time bounded curvature solution, g(t)converges in C∞loc to g(T ) as t → T−, this forces g(T ) to be U(n)-invariant. ByTheorem 3.1.3, g(t) has non-negative bisectional curvature for all t ∈ [0,T ]. Byapplying the short time existence theorem and uniqueness theorem to g(T ), weconclude that the U(n) symmetry is also preserved beyond time T , a contradiction.Case 2: When g0 is not a cigar.In this case, either ξ (∞) = β < 1 or ξ (∞) = 1 with∫ ∞11−ξt dt = ∞. We firstconstruct background metrics for both cases:Proposition 3.3.1. Let g be a smooth U(n)-invariant metric with bounded cur-vature generated by ξ and g is not a cigar. Then given any ε > 0 there exists g˜satisfying(a) The curvature of g˜ is bounded by a constant independent of ε .(b) (1/ε)g˜≤ g≤Cg˜ for some constant C.Proof. Suppose first that ξ (∞) = β < 1. For each k ≥ 1, consider the linear au-tomorphism of Cn given by φk(z) = z/√k and consider the U(n)-invariant Ka¨hlermetric gk := φ ∗k g on Cn. Consider the functions hk(r),ξk(r) and h(r),ξ (r) etc cor-responding to gk and g. Then for each k ≥ 1 we have1. hk(r) = (1/k)h(r/k)2. ξk(r) = ξ (r/k)383. The curvature of gk is bounded by a constant independent of k because gk isisometric to g0.Nowhk(r)h(r)=1kh( rk )h(r)=1kexp(∫ rrkξ (s)sds) (3.3.1)and thus by 0≤ ξ ≤ β we havek−1 ≤ hk(r)h(r)≤ k(β−1). (3.3.2)By Remark 3.1.1, for any k ≥ 1 we havek(1−β )gk ≤ g≤ kgk. (3.3.3)Thus the Proposition follows in this case by the fact that β < 1.Suppose now∫ ∞11−ξt dt = ∞. Let ε > 0 be given. Let ĝ be any U(n)-invariantnon-negative bisectional curvature metric with ĥ(0) = 1 and generated by someξ̂ with ξ̂ (r) = 1 for r ≥ 1. Let the curvature of ĝ be bounded by K̂. For eachk ≥ 1 define the pullbacks ĝk := φ ∗k (ĝ) as before. Let ĥk(r), ξ̂k(r) and ĥ(r) etccorresponding to ĝk and ĝ. Then properties (1) and (2) (3) above still hold, butwith h,ξ ,hk,ξk replaced with ĥ, ξ̂ , ĥk, ξ̂k.STEP 1: First note that by (3.3.1) (applied to hˆ(r)) and the fact that ξˆ ≤ 1, wesee that hk(r) is non-increasing in k for all r. Now fix ε > 0, by∫ ∞11−ξt dt =∞ thereis r0 > 0 such that if r ≥ r0, then for k ≥ 1h(r) =ĥ(r)h(r)ĥ(r)≥ĥk(r)exp(∫ r0ξ̂ (s)−ξ (s)sds)≥1εĥk(r)where we have used the fact that ξ̂ (r) = 1 for r ≥ 1. On the other hand, by (3.3.1)one can see this is true for r ≤ r0 if k is large enough depending on r0. Hence by39Remark 3.1.1 we can find k > 1 depending on ε such that,1εgˆk ≤ gon Cn.STEP 2: Defineξ˜k(r) := ξˆk(r)+ok(r)where ok(r) : [0,∞)→ R is a non-positive smooth function with |ok| ≤ 1k to bechosen. Let h˜k(0) = 1/k and consider the corresponding metric g˜k.Claim 1: There exists a constant Rk > k and a smooth function ok(r) which is0 on [0,Rk] and satisfies:|o′k(r)| ≤4krand|∫ rRkξ˜k(s)−ξ (s)sds|=∣∣∣∣∫ rRk 1+ok(s)−ξ (s)s ds∣∣∣∣≤ 1+2log2 (3.3.4)for r ≥ Rk.We may choose Rk > k such that 1− 1/k ≤ ξ (r) ≤ 1 on [Rk,∞). The con-struction of ok(r) follows from the construction in the proof of Proposition 3.2.1.We first choose a smooth non-increasing function ρ(r) : [0,∞)→ R with ρ = 0 on[0,1], ρ = 1/k on [2,∞) and 0≤ ρ ′ ≤ 2/k. LetI(r) :=∫ r2Rk1+ok(r)−ξ (s)sds.(note that ξ˜k(r) = 1+ ok(r) for r ≥ Rk). For any positive sequence {ri} such thatr0 := Rk and 2ri < ri+1, define ok(r) := 0 if r ∈ [0,r0), ok(r) := ρ(r/r0) if r ∈[r0,2r0), and40ok(r) := 1/k if r ∈ [2r0,r1]= 1/k−2ρ(r/r1) if r ∈ [r1,2r1]=−1/k if r ∈ [2r1,r2]=−1/k+2ρ(r/r2) if r ∈ [r2,2r2]= 1/k if r ∈ [2r2,r3]= 1/k−2ρ(r/r3) if r ∈ [r3,2r3]...(3.3.5)Now for r ∈ [2r0,∞) we have 1− 1/k ≤ ξ (r) ≤ 1, and as long as k ≥ 2 we have|1+ ok(r)− ξ (r)| ≤ 1 as well. The definition of I(r) then gives the following forall i≥ 0I(r) = I(2ri)+∫ r2ri(1+(−1)i/k)−ξ (s)sdsfor r ∈ [2ri,ri+1),I(ri+1)− log2≤ I(r)≤ I(ri+1)+ log2for r ∈ [ri+1,2ri+1).By the fact ξ (r)→ 1, we may choose the ri’s to be the smallest numbers withI(r1) = 1, I(r2) =−1, I(r3) = 1,..etc, and the estimates above give−1− log2≤ I(r)≤ 1+ log2 (3.3.6)for all r ∈ [2Rk,∞). The integral bound in the claim follows from (2.7) and the factthat |1+ok(r)−ξ (r)| ≤ 1 for r ∈ [Rk,2Rk].Finally, we also have |o′k(r)|= 2riρ ′( rri )≤ 4kri ≤ 4kr for all r ∈ [ri,2ri].Claim 2: Let ok(r) be as in Claim 1. Then (1/4eε)g˜k ≤ g≤Ckg˜k for some Ckand the curvature of g˜k is bounded depending only on ĝ.To prove the first part of the claim, when r ≤ Rk we have g˜k(r) = gk(r), and so41we only have to consider when r ≥ Rk. In this case, we have Ck ≥ h(r)/h˜k(r) =(h(Rk)/h˜k(Rk))e∫ rRk1+ok(s)−ξ (s)s ds ≥ 12eε for some Ck where we have used Step 1 andClaim 1.To prove the second part of the claim, note that for r ≥ Rk we have |ξ˜ ′k(r)| =|o′k(r)| ≤ 4/kr andh˜k(r) = h˜k(1)exp(−∫ r1ξ˜k(s)sds)= h˜k(1)exp(−∫ Rk1ξ˜k(s)sds−∫ rRkξ˜k(s)sds)≥ h˜k(1) 1Rk exp(−∫ rRk(ξ˜k(s)−ξ (s)s+ξ (s)−1s+1s)ds)≥ h˜k(1) 1RkRk4e1+δ r= hk(1)14e1+δ r≥ ĥ(1)k14e1+δ r(3.3.7)where in the third line we have used that 0≤ ξ˜k ≤ 1 by definition, and in the fourthline we have used Claim 1 and∫ ∞11−ξt dt = ∞, so that∫ rRk1−ξ (s)s ds≥−δ .Thus |ξ˜ ′k(r)/h˜k(r)| ≤ 16e1+δ/ĥ(1) for r ≥ Rk. Since g˜k(r) = gk(r) for r ≤ Rk,by the previous computations of ξk and hk we have |ξ˜ ′k(r)/h˜k(r)|= |ξ ′k(r)/hk(r)|=|ξˆ ′( rk )/hˆ( rk )|. As ξˆ (r) = 1 if r ≥ 1, so |ξ˜ ′k(r)/h˜k(r)| is bounded unformly indepen-dent of k for r ≤ Rk . As a result, we conclude that the curvature of g˜k is boundedby a constant independent of k by Lemma 3.1.1.Proof of Case 2. By Proposition 3.3.1, for all ε > 0, there is a complete U(n)-invariant metric ĝε with curvature bounded by K with K being independent of εsuch that1εĝε ≤ g0 ≤ c(ε)ĝεfor some constant c(ε) which may depend on ε . By Theorem 2.3.2, there exists42a Ka¨hler-Ricci flow g(t) on [0,Tε), where Tε = 12nKε , such that g(t) is uniformlyequivalent to g0 for all t ∈ [0,T ′) for any T ′ < Tε and has uniformly boundedcurvature on (δ ,T ′) for all 0 < δ < T ′ < Tε . Let ε → 0 and use Theorem 2.4.1,one may conclude the theorem is true. Since g(t) is a bounded curvature solution,the U(n) symmetry and the non-negativity of bisectional curvature can be provedusing the same argument as in Case 1.3.4 Bounding the scalar curvatureStarting from g0 which is U(n)-invariant with non-negative bisectional curvature,we have proved the long time existence of Ka¨hler-Ricci flow in the last section, butbefore we prove the convergence of the flow, we will prove that the scalar curvatureis uniformly bounded on compact subset along the flow, this is an important prop-erty that will be used when proving the convergence. First we need the followinglocalized version of Ni-Tam’s estimate [24, Theorem 2.1].Lemma 3.4.1. Let (Mn,g(t)) be complete non-compact solution of the Ka¨hler-Ricci flow (1.0.1) on M× [0,T ) with bounded nonnegative bisectional curvature.LetF(x, t) = log(det(gi j¯(x, t))det(gi j¯(x,0)))and for any ρ > 0, letm(ρ,x, t) = infy∈B0(x,ρ)F(y, t). Then there is c> 0 dependingonly on n such that for any x0 ∈M and for all ρ, t > 0−F(x0, t)≤c[(1+t(1−m(ρ,x0, t))ρ2)∫ 2ρ0sk(x0,s)ds− tm(ρ,x0, t))(1−m(ρ,x0, t))ρ2] (3.4.1)where B0(x,ρ) is the geodesic ball with respect to g0 = g(0), andk(x,s) :=∫B(x,s)R0(y)dV0is the average of the scalar curvature R0 of g0 over B0(x,s).43Proof. By [24, (2.6)], if Gρ is the positive Green’s function on B0(x0,ρ) withDirichlet boundary value, then∫B0(x0,ρ)Gρ(x0,y)(1− eF(y,t))dV0≤ t∫B0(x0,ρ)Gρ(x0,y)R0(y)dV0+∫ t0∫B0(x0,ρ)Gρ(x0,y)∆0(−F(y,s))dV0ds=: I+ II.(3.4.2)As in [24, p.126],II ≤−tm(ρ,x0, t). (3.4.3)As in [24, (2.8)] there is a constant c1 depending only on n such thatρ2∫B0(x0, 15ρ)(−F(y, t))dV0≤ c1t(1−m(ρ,x0, t))(∫B0(x0,ρ)Gρ(x0,y)R0(y)dV0−m(ρ,x0, t)).Using the fact that ∆0(−F)≥−R0 and [24, Lemma 2.1], we obtain−F(x0, t)≤∫B0(x0, 15ρ)Gρ(x0,y)R0(y)dV0+ c2ρ−2t(1−m(ρ,x0, t))(∫B0(x0,ρ)Gρ(x0,y)R0(y)dV0−m(ρ,x0, t)) (3.4.4)for some c2 depending only on n. As in [24, p.127], we get the result.We also need the following:Lemma 3.4.2. Let g(t) be the complete U(n)-invariant solution of the Ka¨hler-Ricciflow (1.0.1) with nonnegative bisectional curvature. LetF(r, t) := F(z, t) = log(det(gi j¯(z, t))det(gi j¯(z,0)))44where r = |z|2. Then for r ≥ 1 and for all tF(r, t)≥−c−n logr+F(1, t)for some constant c > 0 depending only on g(0). If in addition the generatingfunction ξ of g0 satisfies:limr→∞∫ r11−ξ (s)sds = b < ∞, (3.4.5)then for r ≥ 1 and for all tF(r, t)≥−c+nF(1, t)for some constant c > 0 depending only on g(0).Proof. Consider the functions ξ (r, t),h(r, t), f (r, t) corresponding g(r, t). Then 0≤ξ (r, t) ≤ 1 since g(r, t) has non-negative bisectional curvature, and by Theorem3.1.1 we then get 0≤ h(r, t), f (r, t)≤ 1. Thus for r ≥ 1 we havef (r, t) =1r∫ r0h(s, t)ds≥ 1r∫ 10h(s, t)dt =1rf (1, t).h(r, t) = h(1, t)exp(∫ r1−ξ (s)sds)≥ 1rh(1, t),and using the formula det(gi j¯(r, t)) = h(r, t) fn−1(r, t) we then getdet(gi j¯(r, t))det(gi j¯(r,0))≥det(gi j¯(r, t))≥ 1rnh(1, t) f n−1(1, t)=1rndet(gi j¯(1, t))=1rndet(gi j¯(1, t))det(gi j¯(1,0))·det(gi j¯(1,0)).From this, it is easy to see the first result follows. Now suppose the generating45function ξ of g0 also satisfies (3.4.5). Then for r ≥ 1,h(r, t)h(r,0)=h(1, t)exp(−∫ r1 ξ (s,t)s ds)h(1,0)exp(−∫ r1 ξ (s,0)s ds)≥ h(1, t)h(1,0)exp(∫ r1ξ (s,0)−1sds)≥c1 h(1, t)h(1,0)for some constant c1 > 0 independent of r, t, provided r ≥ 1. Also for r ≥ 1f (r, t)f (r,0)=∫ r0 h(s, t)ds∫ r0 h(s,0)ds=∫ 10 h(s, t)ds+∫ r1 h(s, t)ds∫ 10 h(s,0)ds+∫ r1 h(s,0)ds≥h(1, t)+ c1h(1,t)h(1,0)∫ r1 h(s,0)ds∫ 10 h(s,0)ds+∫ r1 h(s,0)ds≥c2 h(1, t)h(1,0)for some constant c2 > 0 independent of r, t, provided r ≥ 1.Hence for r ≥ 1, at the point |z|2 = rdet(gi j¯(z, t))det(gi j¯(z,0))=h(r, t) f n−1(r, t)h(r,0) f n−1(r,0)≥c3(h(1, t)h(1,0))n≥c3(h(1, t) f n−1(1, t)h(1,0) f n−1(1,0))n=c3(det(gi j¯(1, t))det(gi j¯(1,0)))nfor some c3 > 0, where we have used the fact that f (r, t) ≤ f (r,0). From this thesecond result follows.And now we prove a lemma about bounding the scalar curvature on compact46subsets.Lemma 3.4.3. Let g(t) be as in Theorem 3.3.1. Then for any α > 0 the curvatureof g(t) is uniformly bounded in D(α)× [0,∞), where D(α) = {|z|2 < α}.Proof. Letk(z,ρ) =1V0(z,ρ)∫B0(z,ρ)R(0)be the average of the scalar curvature R(0) of g0 over the geodesic ball B0(z,ρ)with respect to g0. Letk(ρ) = sup|z|≤1k(z,ρ).By [33, Theorem 7], there is a constant c1 such thatk(ρ)≤ c11+ρ. (3.4.6)Suppose |z|2 = r, then the distance ρ(z) = ρ(r) from z to the origin satisfiesρ(z) = ρ(r) =12∫ r0√h√sds≥ c2 logr (3.4.7)for some constant c2 > 0 for all r ≥ 1. LetF(z, t) = logdet(gi j¯(z, t))det(gi j¯(z,0))and let m(ρ, t) = infz∈Cn,ρ(z)≤ρ F(z, t). Fix r0 > 1 and let ρ0 = ρ(r0). Denote−m(ρ0, t) by η(t). By Lemmas 3.4.1 and 3.4.2, there exist positive constants c3,c4independent of t and ρ such thatη(t)≤c4[(1+t(1−m(ρ+ρ0, t))ρ2)K(ρ)− tm(ρ+ρ0, t)(1−m(ρ+ρ0, t))ρ2]≤c4[(1+t(1+ c1+ log r˜(ρ)+η(t))ρ2)K(ρ)+t(c3+ log r˜(ρ)+η(t))(1+ c1+ log r˜(ρ)+η(t)ρ2]47whereK(ρ) =∫ 2ρ0sk(s)dsand r˜(ρ) is such that,ρ+ρ0 =12∫ r˜(ρ)0√h√sds.By (3.4.6) and (3.4.7), there is a constant c5 independent of ρ and t, such that1+η(t)≤ c5(ρ+t(1+η(t))ρ+t(1+η(t))2ρ2+ t).Let ρ2 = 2c5t(1+η(t)), then1+η(t)≤ c6(t12 (1+η(t))12 + t)for some constant c6 > 0 independent of t. From this we conclude that η(t) ≤c7(1+ t) for some constant c7 independent of t. Hence−F(z, t)≤ c7(1+ t)for all z with |z|2 ≤ r0, which implies∫ 2ttR(z,s)ds≤∫ 2t0R(z,s)ds =−F(z, t)≤ c7(1+ t).On the other hand, by the Li-Yau-Hamilton inequality [5], sR(z,s) ≥ tR(z, t) fors≥ t. Hence we haveR(z, t)≤ c8for all t and for all z with |z|2 ≤ r0. This completes the proof of the lemma.3.5 Convergence after rescalingTheorem 3.5.1. Let g(t) be a complete longtime U(n)-invariant solution of theKa¨hler-Ricci flow (1.0.1) with bounded non-negative bisectional curvature, andassume g(0) also has bounded curvature. Then g(t) converges, after rescaling atthe origin, to the standard Euclidean metric on Cn.48In order to prove the theorem, we need the following lemmas.Lemma 3.5.1. Let g(t) be as in Theorem 3.5.1. Suppose the curvature of g(t) isuniformly bounded by c1, in D(α)× [0,∞), where D(α) = {|z|2 < α}. Then thereis a constant c2 depending only on c1 and α such thath(r, t)≤ h(0, t)≤ c2h(r, t); f (r, t)≤ f (0, t)≤ c2 f (r, t) (3.5.1)for all 0 < r < α and for all t.Proof. By Remark 3.1.1 and the fact that g(t) has nonnegative bisectional curva-ture, we have h(0, t)≥ h(r, t) and f (0, t)≥ f (r, t) for all r > 0. On the other hand,we have A(r, t),B(r, t),C(r, t)≤ c1 in D(α)× [0,∞) by hypothesis. Thus C =−2 frf 2gives1f (r, t)− 1f (0, t)≤ c1α2in D(α)× [0,∞), and by f (0, t)≤ f (0,0) = h(0,0) = 1 we getf (0, t)≤(c1α2+1)f (r, t).Also, A = ξr(r,t)h(r,t) ≤ c1 givesξr(r, t)≤ c1h(r, t)≤ c1h(r,0)≤ c1h(0,0) = c1in D(α)× [0,∞), and thus ξ (r, t)≤ c1r, givingh(r, t) = h(0, t)exp(∫ r0−ξ (s, t)sds)≥ exp(−c1α)h(0, t).This completes the proof of the lemma.Proof of Theorem 3.5.1. Let a(t)= h(0, t). We claim that the curvature of 1a(t)g(x, t)converges to 0 uniformly on compact sets. Note that 1a(t)g(x, t) has nonnegativebisectional curvature. Let R(z, t) be the scalar curvature of g(t) at z ∈ Cn. Sup-pose first that limt→∞R(0, t) = 0. Then by the Li-Yau-Hamilton inequality [5], weconclude that limt→∞R(z, t) = 0 uniform on compact sets. Since a(t) ≤ a(0) =h(0,0) = 1, the claim is true in this case.49Suppose on the other hand that there exist k→∞ and c1 > 0 such that R(0, tk)≥c1 for all k. We may assume that tk+1 ≥ tk +1. By the Li-Yau-Hamilton inequalityagain, there is c2 > 0 such that R(0, tk + s)≥ c2 for all k and for all 0≤ s≤ 1. Bythe U(n) symmetry, the Ricci tensor of g(t) at the origin is Ric = Rn g, using theKa¨hler-Ricci flow equation, we haveh(0, tk+1)≤ h(0, tk +1)≤ e−c3h(0, tk)for some c3 > 0 for all k. Hence h(0, tk)→ 0 as k→∞. Since h(0, t) is nonincreas-ing, we have limt→∞ a(t) = limt→∞ h(0, t) = 0. On the other hand, the curvature ofg(t) is uniformly bounded on compact sets by Lemma 3.4.3. Thus our claim is truein this case as well.Consider any sequence tk→ ∞. Let ak = h(0, tk) and let g˜k(x, t) = 1ak g(x,akt +tk). Then g˜k(t) is a U(n)-invariant solution to the Ka¨hler-Ricci flow onCn×[− tkak ,∞).Note that −tk/ak ≤ −tk because ak ≤ 1. By Lemmas 3.5.1, 3.4.3, for any α > 0,g˜k(x,0) is uniformly equivalent to the standard Euclidean metric ge on D(α) (withrespect to k). By the claim above and the Li-Yau-Hamilton inequality [5], thecurvature of the metrics g˜k(x, t) approach zero uniformly (with respect to k) oncompact subsets of Cn× (−∞,0].In particular, we conclude that g˜k(t) is uniformly equivalent to ge in D(α)provided −1≤ t ≤ 0, and thus by Theorem A.1.1, we have for any m≥ 0, there isa c4 depending on α such that|∇me g˜k(0)| ≤ c4on D(α2 ), where ∇e is the derivative with respect to the standard Euclidean metric.From this it is easy to conclude the subsequence convergence of g˜k(0) uniformlyand smoothly on compact subsets of Cn to a flat U(n)-invariant Ka¨hler metric g∞,generated by some ξ∞ say. Since the curvature is zero, we have ξ ′∞ ≡ 0 and thusξ∞ ≡ 0. Moreover, at the origin (g∞)i j¯ = δi j. Hence h∞(0) = 1 which impliesthat (g∞)i j¯ = δi j everywhere. From this the Theorem follows as tk was chosenarbitrarily.50Chapter 4Quasi-projective manifolds4.1 IntroductionIn this chapter, we discuss the Ka¨hler-Ricci flow on quasi-projective manifolds. Acomplex manifold M is said to be quasi-projective if M = M¯ \D for some compactKa¨hler manifold M¯ and D ⊂ M¯ is a divisor with simple normal crossing. Withthis structure, we can define different notions of singularity for Ka¨hler metrics onM. We are interested in cusplike merics on M, which are metrics equivalent to thestandard complete local modelidz1∧dz1¯|z1|2 log2 |z1|2 + in∑j=2dz j ∧dz j¯,where (zi) is a local holomorphic coordinate in which D = {z1 = 0}. As an exam-ple, let η be a Ka¨hler form on M¯ and let S be a holomorphic section of OD thatvanishes precisely on D, a Ka¨hler metric ωˆ on M is said to be a Carlson-Griffithsform ifωˆ = η− i∂ ∂¯ log log2 ||S||2h= η−2i∂ ∂¯ log ||S||2hlog ||S||2h+2i∂ log ||S||2hlog ||S||2h∧ ∂¯ log ||S||2hlog ||S||2h(4.1.1)51for some Hermitian metric h on OD. They are cusplike metrices inroduced in [3],below are some facts about Carlson-Griffiths forms and we refer to [3] and [17] formore details and explanations:1. ωˆ has bounded geometry of infinite order.2. − log log2 ‖S‖2h is bounded above and in L1(M).3. log ωˆn‖S‖2h log2 ‖S‖2hΩ is bounded on M where Ω is any smooth volume form onM.In particular (2) implies that ωˆ is a well defined current on M. (see for example[23] (§8, example 8.15).In [23] the authors showed that if ω0 is cusplike with so called superstandardspatial asymptotics at D, then a bounded curvature cusplike solution ω(t) to (1.0.1)exists on M× [0,T[ω0]), having the same asymptotics for all t, whereT[ω0] := sup{T : [η ]+T (c1(KM)+ c1(OD)) ∈KM} (4.1.2)and KM is the Ka¨hler cone of M. They also showed under a weaker conditioncalled standard spatial asymptotics, a similar bounded curvature solution existson M× [0,T ), for some maximal T where T ≤ T[ω0]. A main point here is thatthe maximal existence time T[ω0] depends only on the cohomology class of theinitial form ω0. One example of metric having superstandard spatial asymptotics isη− i∂ ∂¯ log log2 |S|2+ i∂ ∂¯ϕ where ϕ ∈C∞(M) and these metrics will play a crucialrole in our discussion.In this chapter, we will construct a solution to (1.0.1) whenω0 = η− i∂ ∂¯ log log2 |S|2+ i∂ ∂¯ϕfor general class of potentials ϕ . Assuming that ω0 is only bounded below onM by a cusplike metric, and has zero Lelong number (see the Appendix A.2 forthe definition and the related propostions we are going to use in this chapter), wewill construct a solution to (1.0.1) on M× [0,T[ω0]) which is bounded below by acusplike metric on some definite positive time subinterval of [0,T[ω0]) (see Theorem524.1.2 and below for details). In particular, ω0 may have unbounded curvature onM here. On the other hand, in cases when ω0 may be incomplete on M, includingwhen ω0 is smooth on M, we can still construct solutions on M× [0,T[ω0]) which iscusplike on M for all positive times (see Theorem 4.1.1). In these casesω0 becomesinstantaneously complete on M under (1.0.1). We now describe our results in moredetails below.We first consider the case ω0 ≥ cη on M for some c > 0 where ϕ0 is boundedand smooth on M. In particular, ω0 is typically incomplete on M here. Our mainresult here isTheorem 4.1.1. Let ϕ0 ∈ L∞(M)⋂C∞(M)⋂Psh(M,η) such thatω0 = η+ i∂ ∂¯ϕ0 ≥ cη (4.1.3)for some constant c> 0. Let T[ω0] be as in (4.1.2). Then (1.0.1) has a unique smoothsolution ω(t) on M× [0,T[ω0]) wherec1(t)ωˆ ≤ ωt ≤ c2(t)ωˆ (4.1.4)for all t ∈ (0,T[ω0]) ,and some positive functions ci(t) and Carlson-Griffiths formon ωˆ on M.Also, for any Hermitian metric h on OD and volume form Ω on M, (4.1.6)and (4.1.7) hold on M× [0,T[ω0]) for some ϕ(t) which is bounded on M for eacht ∈ [0,T[ω0]).Remark 4.1.1. Theorem 4.1.1 includes as a special case, when ω0 has conical sin-gularities at D or is in fact smooth on M.Next we consider when ω0 is a complete metric on M in which case ϕ0 may beunbounded on M. Our first result here isTheorem 4.1.2. Let ϕ0 ∈C∞(M)⋂Psh(M,η) have zero Lelong number such thatω0 = η+ i∂ ∂¯ϕ0 ≥ cωˆfor some c > 0 and Carlson-Griffiths form ωˆ on M. Let T[ω0] be as in (4.1.2). Then53the Ka¨hler -Ricci flow (1.0.1) has a smooth solution ω(t) on M× [0,T[ω0]) andω(t)≥ (1n− 4Kˆtc)ωˆ (4.1.5)for all t ≤ c4nKˆwhere Kˆ is a non-negative upper bound on the bisectional curvaturesof ωˆ . Moreover,(1) For any hermitian metric h on OD and volume form Ω on M, (4.1.6) and(4.1.7) hold on M× [0,T[ω0]) where ϕ(t) ≤ c(t) on M× [0,T[ω0]) for somecontinuous function c(t).(2) Suppose further that ω0 is cusplike and−C log log2 ‖S‖2h≤ ϕ0 on M for someconstant C > 0 and Hermitian metric h. Then for any 0 < T < T[ω0], (4.1.6)and (4.1.7) hold on M× [0,T ] where −c log log2 ‖S‖2h′ ≤ ϕ(t) ≤ c on M×[0,T ] for some constant c > 0 and Hermitian metric h′.Remark 4.1.2. It can be proved that there is a unique solution g(t) satisfying (4.1.5)and (1) on the time interval [0, c4nKˆ), though it is not known if our solution is uniqueon the whole time interval [0,T[ω0]).Remark 4.1.3. In Theorem 4.1.1 and 4.1.2 above, we only considered the case ofa single smooth divisor D ⊂M. On the other hand, straight forward extensions ofour definitions and techniques allow us also to consider the case of some collectionof simple normal crossing divisors D1, ..,Dk in which case we can have similarstatements.Note also that Theorem 4.1.2 leaves open the possibility that the solution mayexist beyond t = T[ω0]. Note also, in Theorem 4.1.2 ω0 is complete while the so-lution may not be complete for all positive times. Meanwhile in Theorem 4.1.1,ω0 may be incomplete while the solution is complete for all positive times. Thisseems counterintuitive, and is a result of the stronger a priori estimates in the caseϕ0 is bounded. On the other hand, (2) says cupslikeness is preserved at the poten-tial level for all times in some sense. If we assume ω0 above in fact has boundedcurvature and is sufficiently asymptotic to the standard model at D in a sense, thefollowing Theorem says the solution is indeed cusplike for all times, and [0,Tω) isindeed a maximal time interval.54Theorem 4.1.3. Let η be a smooth Ka¨hler form on M and ωˆ =η−i∂ ∂¯ log log2 ‖S‖2be a Carlson-Griffiths form on M. Let ω0 = ωˆ + i∂ ∂¯ϕ be a smooth completebounded curvature Ka¨hler metric on M such that ϕlog log2 ‖S‖2 → 0,|dϕ|ωˆlog log2 ‖S‖2 → 0and |ω0−ωˆ|ωˆ→ 0 as ‖S‖→ 0. Let T[ω0] be as in (4.1.2). Then (1.0.1) has a uniquesmooth maximal bounded curvature solution ω(t) on M× [0,T[ω0]).In the follow discussion, we will study (1.0.1) through an associated parabolicMonge Ampe`re equation set up as follows. For any Hermitian metric h on OD, andvolume form Ω on M, consider a solution ϕ(t) to the parabolic Monge Ampe`reequation ∂tϕ(t) = log‖S‖2h log2 ‖S‖2h(θt + i∂ ∂¯ϕ(t))nΩ;ϕ(0) = ϕ0.(4.1.6)θt := η+ tχ; χ :=−Ric(Ω)+Θh− i∂ ∂¯ log log2 ‖S‖2hand the associated family of Ka¨hler metricsω(t) := θt + i∂ ∂¯ϕ(t) (4.1.7)on M× [0,T ) for some T . Here Ric(Ω) = −i∂ ∂¯ logΩ and Θh represents the cur-vature form of the metric h on OD.It follows that ω(t) solves (1.0.1) on M× [0,T ), and conversely, if ω(t) solves(1.0.1) on M× [0,T ) then (4.1.7) holds for some solution ϕ(t) to (4.1.6) (see thederivation of (4.2.2)). The equation (4.1.6) is different from the parabolic Monge-Ampe`re equations considered in the earlier works mentioned above in the appear-ance of the ‖S‖2h log‖S‖2h term in the numerator of the right hand side. This termwill be useful in establishing the cusplike-ness of our solutions for positive times.4.2 Proof of Theorem 4.1.1Analogous to [16], we will use Theorem A.2.1 to construct approximation solu-tions to Ka¨hler-Ricci flow from which we will take a limit to obtain the desiredsolution. Our study here is similar in ways to [23] where the authors also studied55cusp type Ka¨hler-Ricci flow solutions. On the other hand, our work here is differentfrom these in the following ways. First, our initial metric is not cusplike (or evencomplete) as in the case of [23], yet we are looking solutions which are cusplikefor positive times. This is one reason that the Monge Ampe`re equation (4.1.6) weconsider is somewhat different than the ones studied in other works. Second, thebackground Carlson-Griffith metric has only L1 volume form so we cannot approx-imate φ0 using the procedure from [29] which is based on Kolodziej’s Lp estimate.In the following we will introduce approximation procedures to overcome thesedifficulties.We first make the following technical assumptions which we will use through-out the rest of the section.Assumption 1. Let η , ωˆ and ϕ0 be as in Theorem 4.1.1 and T[ω0] be as in (4.1.2).Fix some T˜ < T[ω0] and choose a Hermitian metric hˆ on OD and smooth volumeform Ω on M so that ‖S‖2hˆ< 1 on M and1. η+ t(−Ric(Ω)+Θhˆ)> 0 on M× [0, T˜ ]2. η+ t(−Ric(Ω)+Θhˆ+2Θhˆlog‖S‖2hˆ)> 0 on M× [0, T˜ ]Finally, we will abbreviate ‖S‖2hˆand Θhˆ simply by ‖S‖2 and Θ repsectively.We first choose hˆ so that ‖S‖2hˆ< 1 on M. By the definition of T[ω0] we can thenchoose Ω such that (1) holds for t = T˜ . Then, by scaling hˆ smaller if necessary, wemay also assume the inequality in (2) also hold at t = T˜ (by the smoothness of Θon M. The fact that (1) and (2) holds for all t ∈ [0, T˜ ] then follows by interpolationbetween t = 0 and t = T˜ .4.2.1 Proof of Theorem 4.1.1 when ϕ0 ∈C∞(M)In this case, ω0 = η + i∂ ∂¯ϕ0 is a Ka¨hler form on M, from Theorem 8.19 in [23]we haveLemma 4.2.1. For ε > 0 sufficiently small, ωε,0 = ω0 − ε∂ ∂¯ log log2 ‖S‖2h is aCarlson-Griffiths metric on M and (1.0.1) has a bounded curvature solution ωε(t)on M× [0,T[ω0]) with initial data ωε,0.56For ωε(t) above, we will now derive estimates on compact subsets of M×[0,T[ω0]) which are uniform with respect to ε . We will then let ε → 0 to obtaina limit smooth solution ω(t) to (1.0.1) on M × [0,T[ω0]) with initial data ω0 =η+ i∂ ∂¯ϕ0. We may writeωε(t) = θε,t + i∂ ∂¯ϕε(t) (4.2.1)where ϕε(t) solves the parabolic Monge Ampe`re equation on M× [0,T[ω0]): ∂tϕε(t) = log‖S‖2 log‖S‖2(θε,t + i∂ ∂¯ϕε(t))nΩ;ϕε(0) = ϕ0.(4.2.2)θε,t := η+ tχ− ε∂ ∂¯ log log2 ‖S‖2; χ :=−Ric(Ω)+Θ− i∂ ∂¯ log log2 ‖S‖2Indeed, lettingϕε(t) = ϕ0+∫ t0log‖S‖2 log2 ‖S‖2(ωε(t))nΩand defining θε,t as above we see that (4.2.2) is obviously satisfied. On the otherhand, we have θε,0+ i∂ ∂¯ϕε(0) = ωε(0) while on M× [0,T[ω0])(θε,t + i∂ ∂¯ϕε(t))′ =−Ric(Ω)+Θ− i∂ ∂¯ log log2 ‖S‖2+ i∂ ∂¯ (log‖S‖2+ log log2 ‖S‖2+ log(ωnε (t))− logΩ)=i∂ ∂¯ log(ωnε (t))=−Ric(ωε(t))(4.2.3)and it follows from (1.0.1) that (4.2.1) holds on M× [0,T[ω0]). Conversely, reversingthe process above shows that the metric in (4.2.1) satisfies (1.0.1) for any givensolution to (4.2.2).57Note by (2) in Assumption 1 and (4.1.1), for ε > 0 we havec1(t+ ε)ωˆ ≤ θt,ε ≤ c2ωˆ, (4.2.4)for some constants c1,c2 > 0 independent of ε for all t ∈ [0, T˜ ]. We will nowderive estimates for ϕε(t) on M× [0, T˜ ] which will be uniform with respect to εand will yield uniform C∞loc estimates for ωε(t). These will allow us to let ε → 0and T˜ → T[ω0] and obtain a limit solution on M× [0,T[ω0]) satisfying the conditionsin Theorem 4.1.1.We begin with the following C0-estimates:Lemma 4.2.2. On the time interval (0, T˜ ], we have |ϕε | ≤C, |ϕ˙ε | ≤ Ct , where C isindependent of ε sufficiently small.Proof. Fix ε > 0 such that (4.2.4) holds. Since θt,ε is a Carlson-Griffiths form forall t ∈ [0, T˜ ], so the curvature on M =M\D is uniformly bounded. Letψ :=ϕε−Ctfor some C > 0 to be chosen. Then since ωε(t) is a bounded curvature solution, ψis uniformly bounded on M× [0, T˜ ]. We first suppose that ψ attains a maximumvalue on M× [0, T˜ ] at some point (x, t). If t¯ > 0 then using (4.2.2) we have at (x, t)∂tψε ≤ log‖S‖2 log‖S‖2θ nt¯,εΩ−C <−1 (4.2.5)by choosing C sufficiently large independent of ε . Indeed, such a choice of C existsfrom the upper bound in (4.2.4) and property (3) of Carlson-Griffiths metrics (seethe paragraph after (4.1.1)). But (4.2.5) contradicts the maximality of ψ unlesst¯ = 0 in which case ψ ≤ supψ(0) = supϕ0 on M× [0, T˜ ] and thusϕε ≤ supϕ0+CT˜ (4.2.6)on M× [0, T˜ ]. Now in general, suppose ψ does not attain a maximal value onM× [0, T˜ ]. Since ωε(t) is a bounded curvature solution, we have |∂tψε | is boundedon M× [0, T˜ ], and by the Omori-Yau maximum principal we can find a sequencexk ∈M and t¯ ∈ [0, T˜ ] such that ψε(xk, t¯)→ supM×[0,T˜ ]ψε and i∂ ∂¯ψε(xk, t¯)≤ λkθt¯,εwhere λk decreases to 0 as k→ ∞. Combining these with (4.2.2), we may argueas above that for any δ we have ∂tψε(xk, t¯) ≤ −1 for some C independent of ε58and all k sufficeintly large. On the other hand, |∂ 2t ψε |(xk, t¯) is uniformly boundedindependent of k using again that ωε(t) is a bounded curvature solution. These lasttwo facts contradict that ψε(xk, t¯)→ supM×[0,T˜ ]ψε unless t¯ = 0 and we concludeagain as above that (4.2.6) likewise holds on M× [0, T˜ ] in this case as well.Using again that ωε(t) is a bounded curvature solution, we also have |∂ 2t ψε | isbounded on M× [0, T˜ ].By a similar argument, using (4.2.4) and (4.2.2) we may also getinfϕ0+C′∫ t0log(s+ ε)ds≤ ϕε (4.2.7)on M× [0, T˜ ] for some constant C′ > 0 independent of ε . We thus conclude theestimates for |ϕe| in the Lemma hold.Next we will apply the methods in [29] to derive estimates for ϕ˙ε . We have(∂t −∆)ϕ˙ε = trωεχwhere the ∆ denotes the Laplacian with respect to ωε , and also(∂t −∆)(tϕ˙ε −ϕε −nt) =−trωε (η− ε∂ ∂¯ log log2 ‖S‖2)< 0,where the last inequality comes from the fact that η− ε∂ ∂¯ log log2 ‖S‖2h > 0 whenε is small by (4.1.1). Applying the maximum principle in [27], we conclude thesupremum of (tϕ˙e−ϕε−nt) on M×(0, T˜ ], which is indeed finite, is attained whent = 0 and thus ϕ˙ε ≤ ϕε−ϕ0t + n on M× (0, T˜ ]. On the other hand, for sufficientlylarge A independent of ε we have(∂t −∆)(ϕ˙e+Aϕε −n log t)= trωε (χ+Aθt,ε)+A logωnε ‖S‖2 log2 ‖S‖2Ω− (An+ nt)≥ A2trωεθt,ε +A log(t+ ε)nωnεθ nt,ε−Ct≥ A4(θ nt,εωnε)1/n−Ct.(4.2.8)where in the second inequality we have made use of (4.2.4) and property (3) of59Carlson-Griffiths metrics, and in the third inequality we have again used (4.2.4).Now let ψ = (ϕ˙e + Aϕε − n log t), and assume ψ attains a minimum value onM× [0, T˜ ] at some point (x, t). It follows that t¯ > 0, and (4.2.8) then givesωnε (x, t)≥cθ nt,ε(x, t)t¯n for some c > 0 independent of ε . From this, (4.2.4), (4.2.2) and prop-erty (3) of Carlson-Griffiths metrics we may then have ψ ≥ C log t −C and thusϕ˙e ≥ C log t −C on M× (0, T˜ ] for some C > 0 where we have used (4.2.6) and(4.2.7). In general, ψ is only bounded but may not attain a minimum value onM× [0, T˜ ], though we may argue as before, applying the above estimate along anappropriate space-time sequence obtained by the Omori-Yau maximum principle,to conclude ϕ˙e ≥C log t−C on M× (0, T˜ ] for some C > 0 in this case as well. Wethus conclude the estimates for |ϕ˙e| in the Lemma hold.Next we want to derive a Laplacian estimate for ϕe.Lemma 4.2.3. for each t ∈ (0, T˜ ] we have C1(t)ωˆ ≤ωε(t)≤C2(t)ωˆ, for constantsC1(t),C2(t)> 0 independent of ε sufficiently small.Proof. First recall that ωˆ is complete and has uniformly bounded bisectional cur-vature. Then the parabolic version of the Chern-Lu inequality (see [28]) gives:(∂t −∆) log trωε ωˆ ≤Ctrωε ωˆ+Cwhere the constant C depends on the upper bound of the bisectional curvature of ωˆ.For the rest of the proof, C will denote a constant, which may change from line toline, and which is independent of ε . Now by (4.2.4) we may choose A sufficientlylarge independent of ε so that(∂t −∆)(t log trωε ωˆ−Aϕe)≤ trωε (Ctωˆ−Aθt,ε)+ log trωε ωˆ−A logωnε ‖S‖2 log2 ‖S‖2Ω+(An+Ct)≤ − A2trωεθt,ε + log trωεθt,ε +A logθ nt,εωnε−CA log(t+ ε)+C≤ − A4trωεθt,ε −CA log(t+ ε)+C,(4.2.9)Now suppose t log trωε ωˆ −Aϕe attains a maximum value on M× [0, T˜ ] at some60point (x, t). Then if t > 0, using (4.2.9) we have at (x, t) that trωεθt,ε ≤−C log t+Cand sot log trωε ωˆ−Aϕe ≤Ct(log log1t+ log1t+ ε)+C ≤C.In this case it follows thattrωε ωˆ ≤ eCt (4.2.10)on M× [0, T˜ ] where we have used estimate for |ϕε | Lemma 4.2.2. On the otherhand, if t = 0 we also clearly have (4.2.10). In general, In general, t log trωε ωˆ−Aϕeis only bounded but may not attain a maximum value on M× [0, T˜ ], though wemay argue as in the proof of Lemma 4.2.2 and apply the above estimates alongan appropriate sequence in space-time to likewise conclude that (4.2.10) holds onM× [0, T˜ ] in this case as well.Finally, for any t ∈ (0, T˜ ], (4.2.2) givesωnεωˆn= eϕ˙εΩωˆn‖S‖2 log2 ‖S‖2 ≤Ct (4.2.11)on M for some constant Ct > 0 where we have used the estimate for |ϕ˙ε | in Lemma4.2.2 and property (3) of Carlson-Griffiths metrics. The Lemma follows from(4.2.11) and (4.2.10).Completion proof of Theorem 4.1.1 when ϕ0 ∈C∞(M). The previous two lemmasand the Evans-Krylov theory applied to (4.2.2) imply that for any K ⊂⊂ M ands ∈ (0, T˜ ) we havemaxK‖∇kηϕε(t)‖η ≤Ck,s,K,tindependent of ε and t ∈ (s, T˜ ] where the norm and covariant derivative here arewith respect to η . Thus for some subsequence εi → 0, ϕεi will converge locallyuniformly to a smooth solution ϕ to (4.1.6) on M× (0, T˜ ) which is bounded on Mfor each t. Thus ωεi(t) converges locally uniformly to a smooth solution ω(t) tothe flow in equation (1.0.1) on M×(0, T˜ ) and as T˜ < T[ω0] was arbitrary, we may infact assume the convergence to a solution on M× (0,T[ω0]) satisfying the estimatesin Lemma 4.2.3.We now show the limit solution ω(t) in fact converges smoothly uniformly on61compact subsets of M to ω0 as t→ 0, and thus can be extended to a smooth solutionto (1.0.1) on M× [0,T[ω0])with initial dataω0. This basically follows from applyingTheorem 2.3.1 to the sequence ωεi(t), and observing that the completeness of thebackground metric gˆ in that Theorem is in fact not necessary in our case. Wedescribe this in more detail as follows. Consider the family of solutions ωεi(t) onM× [0,T[ω0]). For each i we have ωεi(0)≥ cη on M for some c > 0 independent ofi and we conclude from the proof of Lemma 2.2.1 thatωεi(t)≥ cη (4.2.12)on M× [0,T ] for some c,T > 0 independent of i. For this simply observe that inLemma 2.2.1, the completeness of gˆ is never actually used in the proof. Next wechoose any smooth non-negative function ψ : [0,∞)→ R which is identically zeroin some neighborhood of 0 and let ϕ(t) := ψ(‖S‖2). Then using (4.2.12), as in theproof of Lemma 2.2.3 we may have ‖∇εi,tϕ(t)‖,‖∆εi,tϕ(t)‖ ≤ C on M where thenorms here are relative to ωεi(t) and C is independent of i and t ∈ [0,T ), and by thesame proof there we may conclude that ϕ(t)Rεi(t) ≥C on M× [0,T ] where Rε(t)is the scalar curvature of ωεi(t) and the constant C is independent of i. From thisand the fact that φ was arbitrarily chosen, we can conclude as in Lemma 2.2.3 thatfor any compact K ⊂⊂M we have the upper boundωεi(t)≤ cη (4.2.13)on K× [0,T ] for some c independent of i. The smooth convergence of ωεi(t) oncompact subsets of M× [0,T ] then follows from (4.2.12), (4.2.13) and the Evans-Krylov theory. Thus the limit solution ω(t) extends smoothly to M× [0,T[ω0]) asclaimed above.To complete the proof of the Theorem in this case, it remains only to prove theuniqueness statement which we do in the next sub-section in Proposition 4.2.1.4.2.2 Proof of Theorem 4.1.1 when ϕ0 ∈ L∞(M)⋂C∞(M)By (4.1.3) we can choose some ε > 0 so that in fact we have ϕ0 ∈Psh(M,(1−ε)η).Then, using Theorem A.2.1 we may choose a sequence {ϕ j}⊂C∞(M)⋂Psh(M,(1−62ε)η) so that ϕ j ↓ ϕ0 pointwise on M and locally smoothly on M. In particular, itfollows that1. |ϕ j| ≤C for all j and some C2. ω j = η+ i∂ ∂¯ϕ j ≥ εη for all j on M.Now for each j we let ω j(t) be the solution to (1.0.1) on M× [0,T[ω0]) withinitial data ω j(0) = η + i∂ ∂¯ϕ j constructed in the previous sub-section. UnderAssumption 1, let ϕ j(t) be the corresponding solution to (4.1.6) on M× [0,T[ω0])also as previously constructed. From Lemma 4.2.2 and (1) it follows that |ϕ j|(t)and |ϕ˙ j(t)| are uniformly bounded on compact subsets of M×(0, T˜ ) independentlyof j where T˜ is from Assumption 1. From Lemma 4.2.3 it further follows thatω j(t) is uniformly equivalent to ωˆ on M, independent of j, on compact intervalsof (0, T˜ ). As T˜ < T[ω0] was arbitrary and by applying the arguments in the lastsub-section separately for each j, we may conclude smooth local estimates forω j(t) on compact subsets of M× [0,T[ω0]) which are independent of j, and thatsome subsequence of ω j(t) converges to a solution ω(t) to (1.0.1) on M× [0,T[ω0])satisfying (4.1.4) for all t ∈ (0,T[ω0]).The proof of Theorem 4.1.1 will be complete once we prove uniqueness of sucha solution which we do in the Proposition below. First, we note that by (4.1.3) eachω j(t) will satisfy the lower bound in (4.2.12) for constants c,T > 0 independentof j and thus the limit solution ω(t) likewise satisfies the lower bound in (4.2.12).We will use this fact in the following proof.Proposition 4.2.1. Let ϕ1(t),ϕ2(t) be two solutions to (4.1.6) on M× [0,T[ω0])with initial data ϕ1(0) = ϕ2(0) ∈ L∞(M)⋂C∞(M). Suppose |ϕ1(t)|, |ϕ2(t)| areboth bounded on M× [0,T ) for every T < T[ω0]. Then ϕ1(t) = ϕ2(t) on M× [0,T ].Proof. We assume without loss of generality that the solutions ω1(t) and ϕ1(t) areas constructed as in the proof of Theorem 4.1.1 so far. Now for any T < T[ω0]we prove that ϕ2 ≤ ϕ1 on M× [0,T ]. Let | · |2 be a Hermitian metric such that‖S‖2 < 1 and let Θ denotes its curvature form. As noted above, ω1(t) satisfies theinequality in (4.2.12) on M× [0,ε] for some constant c,ε > 0. Thus for all a > 0we can find Ca→ 0 as a→ 0 such that log (ω1(t)+aΘ)nω1(t)n <Ca on M× [0,T ]. Consider63ψ = ϕ2−ϕ1 + a log‖S‖2−Cat, since ϕ2−ϕ1 is a bounded function, ψ attains amaximum on M× [0,T ] at some point (x¯, t¯). If t¯ > 0, then at (x¯, t¯) we have0 < ∂tψ = log(ω1(t)+aΘ+ i∂ ∂¯ψ)nω1(t)n−Ca < 0,a contradiction. Thus since ψ(x,0) < 0 we have ϕ2 ≤ ϕ1− a log‖S‖2 +Cat, andletting a→ 0 we get ϕ2 ≤ ϕ1.To prove ϕ2 ≥ ϕ1 we argue similarly. Namely, we first choose Ca→ 0 as a→ 0so that log(ω1− aΘ)n/ωn1 ≥Ca. Then we let ψ = ϕ2−ϕ1− a log‖S‖2−C1t andargue as before using the maximum principle that ψ ≥minMψ(0) everywhere thenconclude by letting a→ 0 that ϕ2 ≥ ϕ1 on M.4.3 Proof of Theorem 4.1.2The proof here roughly the same steps as the proof in Theorem 4.1.1. Namely, weconstruct a suitable approximating family for ω0, then consider the correspondingfamily of approximate solutions to (1.0.1) and convergence to a limit solution isproved using the parabolic Monge Ampe`re equation. One major difference hereis that ω0 is complete on M and we want to preserve this property in our approxi-mation. Another major difference is that φ0 is no longer bounded on M in generalwhich again makes the estimates more difficult.We make the following technical assumptions which we will use throughoutthe rest of the section.Assumption 2. Let η , ωˆ,ϕ0, be as in Theorem 4.1.2. Let 0 < T < T˜ < T[ω0] bearbitrary. Choose a Hermitian metric hˆ on OD, a smooth volume form Ω on M¯ anda constant βˆ > 0 such that ‖S‖2hˆ< 1 on M and1. on M× [0, T˜ ] we haveη+ t(−Ric(Ω)+Θhˆ+2Θhˆlog‖S‖2hˆ)> 0642. on M we have(1− cβˆ )η ≤ η− βˆΘhˆ ≤ (1+ cβˆ )ηfor some cβˆ <12 with T < (1− cβˆ )T˜ .3. log log2 ‖S‖2hˆ> 1 on M, and ωˆ = η+ i∂ ∂¯ log log2 ‖S‖2hˆAs before we will abbreviate ‖S‖2hˆand Θhˆ simply by ‖S‖2 and Θ.We first choose some hˆ so that ‖S‖2hˆ< 1 on M. As in the case of Assumption1, we can find a smooth volume form Ω, then scale hˆ smaller if necessary, so thatthe inequality in (1) holds for t = T˜ in which case it must also hold for all t ∈ [0, T˜ ]by interpolation. A choice of βˆ in (2) is justified by the smoothness of Θ on M andby scaling hˆ smaller if necessary. Finally, by scaling hˆ smaller still we may assumethe inequality in (3) holds and that η+ i∂ ∂¯ log log2 ‖S‖2hˆis also a Carlson-Griffithsmetric. Thus without loss of generality we may assume ωˆ = η+ i∂ ∂¯ log log2 ‖S‖2hˆwhere ωˆ is from Theorem 4.1.2.4.3.1 Approximate solutions ωα, j(t)Recall in Theorem 4.1.2 we have ϕ0 ∈C∞(M)⋂Psh(M,η) with zero Lelong num-ber such thatω0 = η+ i∂ ∂¯ϕ0 ≥ 2δωˆ (4.3.1)on M for some δ > 0. We begin by construct a two parameter family ϕα, j ap-proximating ϕ0 as α → 0 and j→ ∞ so that the metrics ωα, j(0) = η + i∂ ∂¯ϕα, jare likewise bounded below for some fixed Carlson-Griffiths metric for all α . Thisuniform lower bound will be key for our later proofs, and this is one reason for ourtwo parameters construction as opposed to a single parameter approximation as inTheorem A.2.1.Lemma 4.3.1. There exists αˆ such that for all 0 < α ≤ αˆ there exists a sequenceϕα, j ∈ Psh(M,η) such that1. ϕα, j decreases to α log‖S‖2 + ϕ0 (as j → ∞) pointwise and smoothly oncompact subsets of M.652. ψα, j = ϕα, j +δ log log2 ‖S‖2 ∈C∞(M)∩Psh(M,(1−δ )η)Proof. Now i∂ ∂¯ log‖S‖2 coincides with a smooth form on M and thus for α > 0sufficiently small we have −δωˆ ≤ αi∂ ∂¯ log‖S‖2 ≤ δωˆ on M and it follows from(4.3.1) that(1−δ )η+ i∂ ∂¯ (α log‖S‖2+δ log log2 ‖S‖2+ϕ0)≥ δωˆ (4.3.2)In particular, since ϕ0 has zero Lelong number (see definition A.2.1) the potentialon the LHS of (4.3.2) approaches −∞ when approaching D givingα log‖S‖2+δ log log2 ‖S‖2+ϕ0 ∈ Psh(M,(1−δ )η) (4.3.3)for all sufficiently small α > 0. Thus by Theorem A.2.1 there existsψα, j ∈C∞(M)∩Psh(M,(1−δ )η) decreasing to α log‖S‖2 +δ log log2 ‖S‖2 +ϕ0 as j→ ∞ point-wise on M and smoothly on compact sets. In particular, we haveη+ i∂ ∂¯ (−δ log log2 ‖S‖2+ψα, j) = δωˆ+(1−δ )η+ i∂ ∂¯ψα, j > δωˆ (4.3.4)so that ϕα, j :=−δ log log2 ‖S‖2+ψα, j ∈ Psh(M,η) decreases toα log‖S‖2+ϕ0 as j→ ∞ pointwise on M and smoothly on compact sets. Thus forall α > 0 sufficiently small ϕα, j satisfies the conclusions in the lemma.Lemma 4.3.2. For each ϕα, j in Lemma 4.3.1, ωα, j(0) = η+ i∂ ∂¯ϕα, j is completewith bounded curvature on M with lower boundωα, j(0)≥ δωˆ (4.3.5)and the Ka¨hler Ricci flow (1.0.1) has a smooth maximal bounded curvature solu-tion ωα, j(t) on M× [0,T[ω0]) with initial condition ωα, j(0) = η + i∂ ∂¯ϕα, j whereT[ω0] is as in definition 4.1.2.Proof. The lower bound in (4.3.5) follows immediately from (4.3.4). On the otherhand we can also write η + i∂ ∂¯ϕα, j = η + i∂ ∂¯ (−δ log log2 ‖S‖2 +ψα, j) where66ψα, j := ϕα, j + δ log log2 ‖S‖2 ∈ C∞(M) as in Lemma 4.3.1 (2), and the Lemmathen follows from Theorem 8.19 in [23] (see also example 8.15).Consider ωα, j(t) as in the above Lemma. In the following we will derive localestimates for ωα, j(t) on M× (0, T˜ ) which will be independent of α, j. These willensure ωα, j(t) converges in C∞loc to a solution ω(t) on M× (0, T˜ ) as α → 0 andj→ ∞. Then we will show that ωα, j(t) in fact converges in C∞loc on M× [0, T˜ ) andthe limit solution is complete for a short time. Since T˜ < T[ω0] in Assumption 2was arbitrary, a diagonal argument will provide a solution on M× [0,T[ω0]) as inTheorem 4.1.2. We may derive as in (4.2.2) thatωα, j(t) = θt + i∂ ∂¯ϕα, j(t) (4.3.6)where ϕα, j(t) solves the parabolic Monge Ampe`re equation: ∂tϕα, j(t) = log‖S‖2 log‖S‖2(θt + i∂ ∂¯ϕα, j(t))nΩ;ϕα, j(0) = ϕα, j.θt := η+ tχ; χ :=−Ric(Ω)+Θ− i∂ ∂¯ log log2 ‖S‖2(4.3.7)on M× [0,T[ω0]).We will derive local uniform bounds of |ϕα, j(t)| and |∂tϕα, j(t)| on K× [ε, T˜ ]where K⊂M is compact and 0< ε are arbitrary where the bounds will independentof α, j. We will use these to derive uniform local trace estimates for ωα, j(t) whichcombined with the local Evans-Krylov estimates will yield the desired uniform C∞locestimates.4.3.2 A priori estimates for ωα, j(t)Recall the choices for 0 < T < T˜ < T[ω0] and hˆ,Ω, βˆ in Assumption 2 and thenotation there. Recall also the definitions of θt and χ from (4.3.7). We fix some αˆfrom Lemma 4.1 and will always assume that α ≤ αˆ in the following.67Local C0 estimates of ϕα, j(t).From Lemma 4.3.1 there is a constant C and constant K(βˆ ) such thatβˆ2log‖S‖2−Kβˆ ≤ ϕα, j <C (4.3.8)on M for all α ≤ βˆ/2 and all j. The upper bound follows simply from Lemma4.3.1 (2) and the fact the ψα, j ∈C∞(M) is decreasing. On the other hand, the factthat ϕα, j ↓ α log‖S‖2 +ϕ0, and ϕ0 has zero Lelong number (see definition A.2.1)and that ‖S‖(x)→ 0 as x→D in M together imply the lower bound in (4.3.8) someconstant Kβˆ > 0 and any α ≤ βˆ/2 and jTheorem 4.3.1. There is a bounded continuous function U(t) on [0, T˜ ] such thatϕα, j(t)≤U(t) on M× [0, T˜ ] for all α and j. There is a continuous function Lβˆ (t)on [0,(1−cβˆ )T˜ ] such that 32 βˆ log‖S‖2+Lβˆ (t)≤ ϕα, j(t) on M× [0,(1−cβˆ )T˜ ] forall α ≤ βˆ/2 and all j.We first proveLemma 4.3.3. We have1. ‖S‖2 log2 ‖S‖2θ ntΩ ≤C1(1+ t) for all t ∈ [0, T˜ ];2. ‖S‖2 log2 ‖S‖2(θt−βˆΘ)nΩ ≥C2t for all t ∈ [0,(1− cβˆ )T˜ ],where the constants Ci > 0 depend on Ω, hˆ and T˜ .Proof. Now it suffices to show that C2t ≤ ‖S‖2 log2 ‖S‖2θ ntΩ ≤C1+C1t for all t ∈ [0, T˜ ],where C is a constant depending only on h and T˜ , since by (2) Assumption 2 wehave for all t ∈ [0,(1− cβˆ )T˜ ]‖S‖2 log2 ‖S‖2(θt − βˆΘ)nΩ≥(1− cβˆ )n‖S‖2 log2 ‖S‖2θ n t1−cβˆΩ≥ 12n‖S‖2 log2 ‖S‖2θ n t1−cβˆΩ.68Now θt = θ˜t + 2ti∂‖S‖2∧∂¯‖S‖2‖S‖4 log2 ‖S‖2 where θ˜t = η− tRicΩ+ tΘ+ 2tΘlog‖S‖2 . Thus θ nt = θ˜ nt +nθ˜ n−1t ∧ (2ti∂‖S‖2∧∂¯‖S‖2‖S‖4 log2 ‖S‖2 ) and‖S‖2 log‖S‖2θ nt = ‖S‖2 log2 ‖S‖2θ˜ nt +nθ˜ n−1t ∧2ti∂‖S‖2∧ ∂¯‖S‖2‖S‖2From this, the fact that 2i∂‖S‖2∧∂¯‖S‖2‖S‖2 is a continuous positive (1,1) form on M¯, andthe positivity of θ˜t on t ∈ [0, T˜ ] we conclude C2t ≤ ‖S‖2 log‖S‖2θ ntΩ ≤ C1 +C1t onM× [0, T˜ ] as claimed.Proof of Theorem 4.3.1. For all α ≤ βˆ/2 and j, considerHε = ϕα, j(t)−∫ t0log[C1(1+ t)]dt− εton M× [0, T˜ ] for any ε > 0 and C1 from Lemma 4.3.3. Since Hε(x,0) = ϕα, j isbounded above by (4.3.8) and |∂tϕα, j(t)| and hence |∂tHε | is bounded on M× [0, T˜ ](by (4.3.7) and that ωα, j(t) is a complete bounded curvature solution to (1.0.1)), itfollows Hε is bounded above on M× [0, T˜ ]. Now suppose Hε attains a maximumvalue on M× [0, T˜ ] at (x¯, t¯). Then if t¯ > 0, using (4.3.7) and Lemma 4.3.3 we haveat (x¯, t¯):∂tHε ≤ log ‖S‖2 log‖S‖2θ nt¯Ω− log[C1(1+ t¯)]− ε ≤−εwhich contradics the maximality assumption. Thus t¯ = 0 in which case we maysimply take U(t) = C+∫ t0 log[C1(1+ t)]dt for some C by (4.3.8). In general, ifHε does not attain a maximum value on M× [0, T˜ ] we may argue as in the proofof Lemma 4.2.2 and apply the above estimates along an appropriate sequence inspace-time (using the Omori-Yau maximum principle) and likewise take U(t) =C+∫ t0 log[C1(1+ t)]dt for some C in this case as well.For the lower bound we take Qε(x, t)=ϕα, j(x, t)− βˆ log |S(x)|2−∫ t0 log(C2t)dt+εt on M× [0,(1− cβˆ )T˜ ] for any ε > 0 and C2 from Lemma 4.3.3. It follows from(4.3.8), (4.3.7) and that ωα, j(t) is a bounded curvature solution that Qε(x, t)→ ∞69uniformly as x approaches D on M × [0, T˜ ] and hence Qε(x, t) attains an inte-rior minimum on M× [0,(1− cβˆ )T˜ ]. Using again Lemma 4.3.3 we may argueas above for the upper bound and conclude that Lβˆ (t) can be taken as −Kβˆ +∫ t0 log(C2t)dt.Local C0 estimates of ϕ˙α, j(t).Theorem 4.3.2. We have ϕ˙α, j(t)≤ U(t)−βˆ log‖S‖2+Kβˆt +n on M× [0,(1− cβˆ )T˜ ] forall α ≤ βˆ/2 and all j.Proof. The proof is the same as Proposition 3.1 in [16]. Let H = tϕ˙α, j(t)−(ϕα, j(t)−ϕα, j)− nt. Then using (4.3.7) we have (∂t −∆)H < 0, where ∆ is theLaplacian with respect to ωα, j(t). Also, since that ωα, j(t) is a bounded curvaturesolution it follows H is a bounded function on M× [0, T˜ ], and thus by the maxi-mum principle in [27] we H ≤ supx∈M H(x,0) = 0 on M× [0, T˜ ]. Then combiningwith Lemma 4.3.3 and (4.3.8) we obtain the theorem.Theorem 4.3.3. For all A > 0 with 0 < T˜ − 1A , there exists a smooth functionF(‖S‖2(x), t) on M× (0,(1− cβˆ )(T˜ − 1A)] such that ϕ˙α, j(x, t) ≥ F(‖S‖2(x), t) onM× (0,(1− cβˆ )(T˜ − 1A)] for all α ≤ 2βˆ and all j.Proof. For all sufficiently small ε , there exists a constant C > 0 such thatθt − sεi∂ ∂¯ log log2 ‖S‖2 ≥Cωˆ (4.3.9)on M× [0, T˜ ] for all 1≤ s≤ 2Fix some A > 0 with 0 < T˜ − 1A and ε as above. Let Q = ϕ˙α, j(t)+A(ϕα, j(t)−βˆ log‖S‖2+ε log log2 ‖S‖2)−n log t. By our previous bounds, Q→∞ on M as t→0 or ‖S‖→ 0. In fact, from (4.3.7) and that ωα, j(t) is a bounded curvature solution,Q(x, t)→ ∞ uniformly on M as x approaches D for all t ∈ [0,(1− cβˆ )(T˜ − 1A)]. SoQ has a minimum on M× (0,(1− cβˆ )(T˜ − 1A)] at some point (x¯, t¯) with t¯ > 0. Let∆ be the Laplacian with respect to ωα, j(t), using (4.3.7), we have70(∂t −∆)(ϕα, j(t)− βˆ log‖S‖2+ ε log log2 ‖S‖2= ϕ˙α, j−n+Trωα, j(θt − βˆΘ− εi∂ ∂¯ log log2 ‖S‖2)(∂t −∆)ϕ˙α, j(t) = Trωα, jχ.Then at (x¯, t¯) we have the following, where we will use C to denote a constantwhich is independent of α , j and which may differ from line to line.0 ≥ (∂t −∆)Q(x¯, t¯)= ATrωα, j(θt¯+ 1A − βˆΘ− εi∂ ∂¯ log log2 ‖S‖2)+Aϕ˙α, j−nA− nt¯≥ A(1− cβˆ )Trωα, j [θ 11−cβˆ (t¯+1A )− 11− cβˆεi∂ ∂¯ log log2 ‖S‖2]+Aϕ˙α, j−nA− nt¯.≥ AC(1− cβˆ )Trωα, j ωˆ+A log‖S‖2 log‖S‖2ωnα, jΩ−nA− nt¯≥ CTrωα, jωˆ +C logωnα, jωˆn−Ct¯.≥ CTrωα, j ωˆ−Ct¯≥ C(ωˆnωnα, j) 1n−Ct¯.where we have used Assumption 2 in the third line, cβˆ ≤ 12 and (4.3.9) in the fourthline, property (3) of Carlson-Griffiths metrics, and the fact 1λ +C logλ is boundedbelow by some constant depending on C in the sixth line. Therefore, at (x¯, t¯),ωnα, j ≥Ct¯nωˆn and so ϕ˙α, j(x¯, t¯)≥C+n log t¯ by (4.3.7) and property (3) of Carlson-Griffiths metrics. Since log log2 ‖S‖2 > 1 by Assumption 2, we have Q(x¯, t¯) ≥C+A(ϕα, j(x¯, t¯)− βˆ log |S(x¯)|2). By Theorem 4.3.1, ϕα, j(t)− βˆ log‖S‖2 ≥C andso Q(x¯, t¯)≥C. From this, and the upper bound of ϕα, j(t) from Theorem 4.3.1, weconclude the lower bound for ϕ˙α, j(t) in the Theorem.71Local trace estimates for ωα, j(t)Note for all α and j, since ωα, j(t) is a bounded curvature solution on M× [0,T[ω0]),so ωα, j(t) will be uniformly equivalent to ωˆ on any closed subinterval of [0,T[ω0]).In particular, Trωˆωα, j(t) will be a bounded function on M× [0,T ]Theorem 4.3.4. There is a smooth function G(‖S‖2(x), t) on M× (0,(1− cβˆ )T˜ ]such that Trωˆωα, j(x, t)≤ G(‖S‖2(x), t) for all 2α ≤ βˆ and all j.Proof. ConsiderQ(·, t) = t logTrωˆωα, j(t)−B(ϕα, j(t)− βˆ log‖S‖2+ ε log log2 ‖S‖2),where ε is chosen as in (4.3.9) and B > 0 is a large constant which will be de-termined later, independently of α, j. Now Q(x,0) → −∞ as x approaches Dfrom (4.9), and from (4.3.7) and that ωα, j(t) is a bounded curvature solution,Q(x, t)→−∞ uniformly as x approaches D for all t ∈ [0,(1−cβˆ )T˜ ]. Hence Q(x, t)attains a maximum on M× [0,(1−cβˆ )T˜ ] at some point (x¯, t¯). In the following, Ci’swill denote positive constants independent of α, j.If t¯ > 0, then 0 ≤ (∂t −∆)Q(x¯, t¯), where ∆ is the Laplacian with respect toωα, j(t). Also, we have(∂t −∆) logTrωˆωα, j(t)≤C1Trωα, j ωˆ (4.3.10)for some constant C1 depending only on ωˆ (see [28]), so(∂t −∆)t logTrωˆωα, j(t) = logTrωˆωα, j(t)+ t(∂t −∆) logTrωˆωα, j(t)≤ logTrωˆωα, j(t)+C1tTrωα, j(t)ωˆ.Using the computations in the proof of Theorem 4.3.3, we have(∂t −∆)(ϕα, j(t)− βˆ log‖S‖2+ ε log log2 ‖S‖2)≥ ϕ˙α, j−n+(1− cβˆ )C2Trωα, j ωˆ.72Therefore at (x¯, t¯), using that cβ < 1/2 from Assumption 2, we have0≤ (∂t −∆)Q≤ logTrωˆωα, j−Bϕ˙α, j +nB+(C1t− 12BC2)Trωα, j ωˆ≤ logTrωˆωα, j−Trωα, j ωˆ−Bϕ˙α, j +nBwhere in the second line we have assumed a choice B, independent of α, j and t¯,such that C1T˜ − 12 BC2 <−1.Since Trωˆωα, j ≤ (Trωα, j ωˆ)n−1ωnα, jωˆn and ϕ˙a, j ≥C3 logωnα, jωˆn for some C3 depend-ing only on h and ωˆ , putting them into the above expression, we get0≤ (n−1) logTrωα, j ωˆ+(1−BC3) logωnα, jωˆn−Trωα, j ωˆ+C4.Assume further that BC3 > 2, we have0 ≤ −Trωα, j ωˆ+(n−1) logTrωα, j ωˆ− logωnα, jωˆn+C4≤ −12Trωα, j ωˆ− logωnα, jωˆn+C4,we used −x+C logx is bounded above for x > 0 by some constant depending onC. Now let λi be the eigenvalue of ωα, j(x¯, t¯) relative to ωˆ(x¯, t¯) and let C denote apositive constant independent of α, j which may differ from line to line. Then theprevious equation says∑i(12λi+ logλi)≤Cand from the fact that the function 1/2x+ logx is bounded below for all x > 0,we get that ( 12λi + logλi) ≤C for each i, thus Trωˆωα, j(x¯, t¯) ≤C. Since ϕα, j(t)−βˆ log‖S‖2≥C and ε log log2 ‖S‖2≥ 0 we conclude Q(x¯, t¯)≤C. Thus by our earlierobserved upper bound for Q(x,0) we get Q(x, t)≤C on M× [0,(1−cβˆ )T˜ ] and theTheorem follows from this and the upper bound in Theorem 4.3.1.4.3.3 Completion of Proof of Theorem 4.1.2Now recall our family ωα, j(t) of solutions to (1.0.1) on M× [0,T[ω0]) from Lemma4.3.2. Recall that we wrote ωα, j(t) = θt + i∂ ∂¯ϕα, j(t) where ϕα, j(t) solves (4.3.7)73on M× [0,T[ω0]). Also recall the choices made in Assumption 2, and in particularthat 0 < T < T˜ < T[ω0] was arbitrary.From the Theorem 4.3.3, 4.3.4 and (4.3.7), for any ε > 0 and compact subsetsK1 ⊂⊂ K2 ⊂⊂M we may haveC1η ≤ ωα, j(t)≤C2η (4.3.11)on K2× [ε,T ] for some constants Ci independent over all α ≤ βˆ/2 and all j. Itfollows from this and the estimates from the Evans-Krylov theory (see also [32] fora maximum principle proof of these for (1.0.1)), that for some αk→ 0, jk→∞, wehave ωαk, jk(t) converges on K1× [ε,T ] smoothly to a limit solution ω(t) to the flowin equation (1.0.1). As T < T[ω0] was chosen arbitrarily, by a diagonal argumentwe may in fact assume ωαk, jk(t) converges on M× (0,T[ω0]), smoothly on compactsubsets, to a limit solution ω(t) to the flow in (1.0.1), while also ωαk, jk(0)→ ω0smoothly on compact subsets of M. By applying Theorem 2.3.1 to the sequenceωαk, jk(t) on M× [0,T[ω0]) and observing the uniform lower bound on ωαk, jk(0) ≥δωˆ in (4.3.5), we see that ωαk, jk(t) actually converges smoothly on M× [0,T[ω0])to a limit solution satisfying (4.1.5). In other words the solution ω(t) extendssmoothly on M× [0,T[ω0]) and satisfies (4.1.5).We now show (1) in the Theorem 4.1.2 is satisfied. Fix any Hermitian metric hon OD and smooth volume form Ω on M. Then as in our derivation of (4.3.7)we see that ϕ(t) := ϕ0 +∫ t0 log log‖S‖2h log‖S‖2h(ω(t))nΩsolves (4.1.6) on M ×[0,T[ω0]) and (4.1.7). In particular, ϕ(t) = limk→∞uαk, jk(t) where uαk, jk(t) = ϕαk, jk +∫ t0 log log‖S‖2h log‖S‖2h(ωαk, jk(t))nΩand uαk, jk(t) solves (4.1.6) on M×[0,T[ω0])withinitial data ϕαk, jk . To see that the upper bound in (1) holds, note that the estimatein Lemma 4.3.3 (1) in fact holds for any, and hence our, choice of h for some con-stant C1. Then from the proof of the upper bound in Theorem 4.3.1, there existsa continuous function U(t) such that uαk, jk(t) ≤ U(t) and hence ϕ(t) ≤ U(t) onM× [0,T[ω0]). This completes the proof of (1) in the Theorem.Finally, we show that (2) holds. Let ϕ0 be as in (2). For any choice of 0 <T < T˜ < T[ω0] and corresponding subsequent choices in Assumption 2, considersolutions ϕα, j(t) to (4.3.7) on M× [0,T ) constructed in the proof of Theorem 4.1.274so far. Now if ϕ0 also satisfies the lower bound in (2) then we may replace theestimates in (4.3.8) withα log‖S‖2−C log log2 ‖S‖2 ≤ ϕα, j <C (4.3.12)for some C and all α ≤ αˆ and all j where αˆ is from Lemma 4.1.Now for αˆ sufficiently small, observe that the estimate in Lemma 4.3.3 (2) stillholds after replacing βˆ with any α ≤ αˆ . Now repeating the proof of the lowerbound in Theorem 4.3.1, but using instead the functionQε(x, t) = ϕα, j(x, t)−2α log‖S(x)‖2−∫ t0log(C2s)ds+ εt,we may haveϕα, j(t)≥−α log‖S‖2−C log log‖S‖2+∫ t0log(C2s)dson M× [0,T ] for all α ≤ αˆ and all j. The a priori estimates we derived previouslyimply that ϕαk, jk(t) converges smoothly on compact subsets of M× [0,T ] to someϕ(t) satisfying the bounds in (2). This completes the proof of Theorem 4.1.2 (2).4.4 Proof of Theorem 4.1.3We begin with the following Theorem from which Theorem 4.1.3 will follow. Inthe following, for any complete Ka¨hler manifold (M,ω) with bounded curvature,we use T (ω) to denote the maximal existence time of a complete bounded curva-ture solution to the Ka¨hler-Ricci flow (1.0.1) starting from ω . Also, we say γ(x)is a distance like function on (M,ω) if for some p ∈ M and C1,C2 > 0 we haveC−11 d(p,x)≤ γ(x)≤C1d(p, ·) whenever d(p,x)>C2, where d(p, ·) is the distancefunction from p on (M,ω). We begin by proving the following Theorem:Theorem 4.4.1. Let (M, ωˆ) be a complete Ka¨hler manifold with bounded cur-vature. Let γ : M → R be a smooth distance-like function with |∇ˆγ|ωˆ < C and|i∂ ∂¯ γ|ωˆ < C for some constant C. Let ϕ ∈ C∞(M) such that |ϕ|/γ → 0 and|∇ˆϕ|ωˆ/γ → 0 as γ → 0. If ω = ωˆ + i∂ ∂¯ϕ is a complete metric with boundedcurvature and satisfies |ω− ωˆ|ωˆ → 0 as γ → 0, then T (ω) = T (ωˆ).75Proof. Let ρ : R→ R be a smooth function such that ρ = 1 on [0,1] and ρ = 0 on[2,∞). Define ρR : M→R by ρR = ρ(γ/R) and let ωR = ωˆ+ i∂ ∂¯ (ρRϕ). We claimthat if R is sufficiently large then ωR is a complete Ka¨hler metric and there existsCR→ 1 as R→ ∞ such that 1CRω ≤ ωR ≤CRω .We have ωR = ρRω + (1− ρR)ωˆ + 2Re(i∂ρR ∧ ∂¯ϕ) + iϕ∂ ∂¯ρR. Since |ω −ωˆ|ωˆ → 0 as γ → ∞, we have 1CRω ≤ ρRω+(1−ρR)ωˆ ≤CRω for some CR→ 1 asR→ ∞. Now it suffices to show that |2Re(i∂ρR∧ ∂¯ϕ)|ωˆ → 0 and |iϕ∂ ∂¯ρR|ωˆ → 0uniformly on M as R→ ∞.For any point in M, we have|2Re(i∂ρR∧ ∂¯ϕ)|ωˆ ≤ |ρ ′( γR)R∂γ ∧ ∂¯ϕ|ωˆ≤ |ρ′( γR)|R|∇ˆγ|ωˆ |∇ˆϕ|ωˆ≤C(maxR|ρ ′|)χγ−1[R,2R]R|∇ϕ|ωˆ≤ 2C(maxR|ρ ′|)χγ−1[R,2R]|∇ϕ|ωˆγ.Because |∇ˆϕ|ωˆ/γ→ 0 as γ→∞, the function on the right hand side convergesuniformly to 0 as R→ ∞. Similar argument works for|iϕ∂ ∂¯ρR|ωˆ = |ϕρ ′( γR)i∂ ∂¯ γR+ϕρ ′′(γR)i∂γ ∧ ∂¯ γR2|ωˆ .Therefore, we have a family of complete Ka¨hler metrics ωR such that 1CRω ≤ωR ≤ CRω with CR → 1 as R→ ∞ and it is clear that ωR has bounded curvature.Therefore, by Theorem 2.4.2, we have 1CR T (ωR)≤ T (ω)≤CRT (ωR). On the otherhand, since ρRϕ has compact support, by Theorem 4.1 in [23], we have T (ωR) =T (ωˆ) for all R. Therefore, passing the limit R→ ∞ we obtain T (ω) = T (ωˆ).Proof of Theorem 4.1.3. The uniqueness of bounded curvature solutions followsfrom [13]. Let p ∈M and let dωˆ(p, ·) be the distance function to p relative to ωˆ .76Let γ(x) := log log2 |S(x)|2 on M. Then from (4.1.1) we may writeωˆ = η¯− i∂ ∂¯ γ(x) = η−2ddc log‖S‖2hlog‖S‖2h+2i∂γ ∧ ∂¯ γ.Noting that η as well as the numerator of the second term above are smooth formson M, we see that for all x ∈M sufficiently close to D, or equivalently when γ(x) issufficiently large, we have C−1γ(x)≤ dωˆ(p,x)≤Cγ(x) and ‖dγ(x)‖ωˆ <C for someconstant C. Moreover, for all x ∈M sufficiently close to D we also see from abovethat −i∂ ∂¯ γ(x) > 0, and from this and the first equality above we may concludethat ‖i∂ ∂¯ γ(x)‖ωˆ ≤C for some C independent of x. In other words, γ satisfies theassumption in Theorem 4.4.1 relative to ωˆ , and Theorem follows immediately.Remark 4.4.1. In Theorem 4.1.3 we can remove the condition on dϕ if we assumeω0 has the same standard spatial asymptotics as that of ωˆ as defined in [23]. Asan example, if ω = ωˆ+ i∂ ∂¯ log loglog2 ‖S‖2 defines a metric, then it has standardspatial asymptotics at D but not superstandard spatial asymptotics (see example8.12 in [23]) while Theorem 4.1.3 still provides a bounded curvature solution onM× [0,T[ω0]).77Bibliography[1] Z. Blocki, S. Kolodziej, On regularization of plurisubharmonic functions onmanifolds, Proc. Amer. Math. Soc. 135(7) (2007): 2089-2093. → pages iii,83, 84, 85[2] E. Cabezas-Rivas, B. Wilking., How to produce a Ricci Flow via Cheeger-Gromoll exhaustion, to appear in J. Eur. Math. Soc., arXiv:1107.0606 (2011).→ pages 5, 30[3] J. Carlson, P. Griffiths , A defect relation for equidimensional holomorphicmappings between algebraic varieties., Ann. Math. 95 (1972), p. 557584 (En-glish). → pages 52[4] H-D. Cao, Deformation of Ka¨hler metrics to Ka¨hler -Einstein metrics on com-pact Ka¨hler manifolds , Invent. Math. 81 (1985), 359 372. → pages 8, 12[5] H.-D. Cao, On Harnack’s inequalities for the Ka¨hler-Ricci flow, Invent. Math.109 (1992), no. 2, 247–263. → pages 48, 49, 50[6] H-D. Cao, Existence of gradient Ka¨hler-Ricci solitons, Elliptic and ParabolicMethods in Geometry, Minnesota, (1994), 1-16. → pages 5, 37[7] H-D. Cao, Limits of solutions to the Ka¨hler-Ricci flow, J. Diff. Geom., 45(1997), 257-272. → pages 5[8] A. Chau, L.-F. Tam, On the complex structure of Ka¨hler manifolds with non-negative curvature, J. Diff. Geom, 73(2006), 491-530. → pages 5[9] A. Chau, K.-F. Li, L. Shen, Ka¨hler-Ricci flow of cusp singularities on quasiprojective varieties, arXiv:1708.02717. → pages vi78[10] A. Chau, K.-F. Li, L.-F. Tam, Deforming complete Hermitian metrics withunbounded curvature, Asian J. Math. 20(2), 267-292, 2016 → pages vi[11] A. Chau, K.-F. Li, L.-F. Tam, An existence time estimate for Ka¨hler-Ricciflow, Bull. London Math. Soc. 48(4), 699-707, 2016 → pages vi[12] A. Chau, K.-F. Li, L.-F. Tam, Longtime existence of the Ka¨hler-Ricci flow onCn , Trans. Amer. Math. Soc., 369(8), 5747-5768, 2017 → pages vi[13] B.-L. Chen, X.P. Zhu, Uniqueness of the Ricci flow on complete noncom-pact manifolds,J. Differential Geom. 48(4), Volume 74, Number 1, 119-154,(2006). → pages iii, 2, 10, 30, 76[14] X. X. Chen, Y. Q. Wang, Bessel functions, Heat kernel and the ConicalKa¨hler-Ricci flow, Journal of Functional Analysis, 269 (2015), 551632 →pages 7[15] J.-P. Demailly, Complex Analytic and Differential Geometry, 1997,http://www-fourier. ujf-grenoble.fr/demailly/books.html. → pages 83, 84, 85[16] V. Guedj, A. Zeriahi, Regularizing properties of the twisted Ka¨hlerRicci flow,J. Reine Angew. Math. (2016) → pages 55, 70[17] H. Guenancia, Ka¨hler-Einstein metrics with mixed Poincar and cone singu-larities along a normal crossing divisor, Ann. Inst. Fourier 64 (3), 1291-1330(2014) → pages 52[18] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. DifferentialGeom. 17 (1982), 255 306. → pages 8[19] P. Klembeck, A complete Ka¨hler metric of positive curvature on Cn, Proc.Amer. Math. Soc. 64 (1977), 313-316. → pages 5[20] M.-C. Lee, L.-F. Tam, Chern-Ricci flows on noncompact complex manifolds,arXiv:1708.00141. → pages 5[21] G. Liu, On Yaus uniformization conjecture, arXiv:1606.08958. → pages 579[22] J. W. Liu, X. Zhang, The Conical Ka¨hler-Ricci Flow with Weak Initial Dataon Fano Manifolds, Int. Math. Res. Notices, 2017(17), 5343-5384 (2017) →pages 7[23] J. Lott, Z. Zhang, Ricci flow on quasi-projective manifolds, Duke Math. J.(2011), 156(1), 87-123 → pages iii, 2, 6, 10, 52, 55, 56, 67, 76, 77[24] L. Ni, L.-F. Tam, Ka¨hler-Ricci flow and the Poincare´-Lelong equation,Comm. Anal. Geom. 12 (2004), no. 1-2, 111–141. → pages 12, 43, 44[25] L. M. Shen, Unnormalize conical Ka¨hler-Ricci flow, arXiv:1411.7284. →pages 7[26] W.,-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diff.Geom. 30.1 (1989), 223-301. → pages 2, 9, 13, 36[27] W.X. Shi, Ricci flow and the uniformization on complete noncompact Ka¨hlermanifolds, J. Diff. Geom. 45.1 (1997), 94-220. → pages 2, 9, 16, 59, 70[28] J. Song, G. Tian, The Ka¨hler-Ricci flow on surfaces of positive Kodaira di-mension, Invent. Math. 170 (2007), no. 3, 609653 → pages 11, 60, 72[29] J. Song, G. Tian, The -Ricci flow through singularities, Invent. Math. 207(2017), no. 2, 519595 → pages 56, 59[30] G. Tian, S.-T. Yau, Complete Ka¨hler manifolds with zero Ricci curvature. I.,J. Amer. Math. Soc. 3 (1990), 579-609. → pages 83[31] G. Tian, Z. Zhang, On the Ka¨hler-Ricci flow on projective manifolds of gen-eral type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179192. → pages8[32] M. Sherman, B. Weinkove, Interior derivative estimates for the Khler-Ricciflow. Pacific Journal of Mathematics, 257(2), 491-501. (2012) → pages 74,82, 83[33] H.-H Wu, F. Zheng, Examples of positively curved complete Ka¨hler mani-fold, Geometry and Analysis Volume I, Advanced Lecture in Mathematics 17,80Higher Education Press and International Press, Beijing and Boston, 2010,pp. 517542 → pages 5, 28, 29, 30, 47[34] B. Yang, On a problem of Yau regarding a higher dimensional generalizationof the Cohn-Vossen inequality, Math. Ann. 355(2) (2013), 765-781. → pages29[35] B. Yang, F. Zheng, U(n)-invariant Ka¨hler-Ricci flow with non-negative cur-vature, Comm. Anal. Geom . 21, (2013) no. 2, 251–294. → pages 5, 3081AppendixA.1 Interior estimatesLet us first fix some notations and terminology. (Mn, gˆ) is said to have boundedgeometry of infinite order if the curvature tensor and all its covariant derivativesare uniformly bounded on M. In particular, the solution g(t) in Theorem 2.1.2 hasbounded geometry of infinite order for t > 0.Also, we will denote the geodesic ball with respect to the metric g with centerat p and radius r by Bg(p,r). The following theorem can be found in [32].Theorem A.1.1. Let (Mn, gˆ) be a complete non-compact Ka¨hler manifold withbounded geometry of infinite order. Let h(t) be a solution of Ka¨hler-Ricci (1.0.1)on M× [0,T ) with initial condition h0 which is a complete Ka¨hler metric. For anyx ∈M, suppose there is a constant N > 0, such thatN−1gˆ≤ h(t)≤ Ngˆ (A.1.1)on Bgˆ(x,1)× [0,T ). Let ∇ˆ be the covariant derivative with respect to gˆ. Then(i)|∇ˆkh|2gˆ ≤Cktkon Bgˆ(x,1/2)× (0,T ), for some constant Ck depending only on k, gˆ, n, Tand N.(ii) If we assume |∇ˆkh0|2gˆ is bounded in Bgˆ(x,1) by ck, for k ≥ 1, then|∇ˆkh|2gˆ ≤Ck,82on Bgˆ(x,1/2)× [0,T ) for some constant Ck depending only on k, ck, n, T andN.Proof. Since gˆ has bounded geometry of infinite order, by [30], for any x∈M thereexists a local biholomorphism φx : D→M, where D=D(1) is the open unit ball inCn, satisfying the following in D(a) φx(0) = x, φx(D)⊂ Bˆ(x,1), φx(D)⊃ Bˆ(x,2δ ) for some δ > 0 which is inde-pendent of x.(b) C−1δi j¯ ≤ (φ ∗x (gˆ))i j¯ ≤Cδi j¯ for some C independent of x.(c)∣∣∣∣∣∂ l(φ ∗x (gˆ))i j¯∂ zL∣∣∣∣∣≤Cl for any l, i, j and multi index L of length l for some con-stant Cl which is independent of x.Consider φ ∗x (h(t)), which clearly will solve (1.0.1) on D(1)× [0,T ). By theEvans-Krylov theory for fully non-linear elliptic and parabolic equations (see also[32] for a maximum principle proof in the case of Ka¨hler Ricci flow), the resultfollows.A.2 Plurisubharmonic functionsLet M be a compact complex manifold with smooth Ka¨hler metric η .Definition A.2.1. ϕ is called plurisubharmonic on the compact Ka¨hler manifold(M,η), written ϕ ∈ Psh(M,η), when ϕ : M → R is upper semi-continuous andbounded above, and for any local holomorphic coordinate domain Uα , ηα +ϕ is aclassical plurisubharmonic function in Uα where ηα is a local Ka¨hler potential. Inthis context ϕ ∈C∞(M)⋂Psh(M,η) is said to have zero Lelong number if for anyc > 0 we havelimd(x,D)→0ϕ(x)c log‖S‖h → 0where d(·,D) is the distance to D⊂M relative to η .Let ϕ ∈ Psh(M,η)) be given. By [15], or [1] for a simpler proof in our setting,there exists a decreasing sequence ϕ j ∈C∞(M)⋂Psh(M,η) converging pointwise83to ϕ . By a slight modification of the proof in [1], we may assume this convergenceactually holds in C∞loc(M) when ϕ ∈ C∞(M). We include the statement and proofof this below for completeness.Theorem A.2.1. Suppose that ϕ ∈ Psh(M,η)∩C∞(M). Then there exists a se-quence ϕ j ∈C∞(M) with ϕ j ↓ ϕ pointwise on M and smoothly uniformly on com-pact subsets of M.Proof. Let S j be an increasing sequence of open sets exhausting M where eachS j is compact. Let m j = min{ϕ(x) : x ∈ S j} and define ϕ j = max{ϕ,m j}+ 1j .Then ϕ j ∈C0(M)⋂C∞(S j)⋂Psh(M,η) with ϕ j ↓ ϕ pointwise on M and smoothlyuniformly on compact subsets. Now for each j, suppose there exists a sequenceϕ j,k ∈ Psh(M,η)∩C∞(M) with ϕ j,k ↓ ϕ j pointwise uniformly on M and smoothlyuniformly on S j−1. Then for any diverging sequence a j, we have ϕ j,a j ↓ ϕ point-wise on M and smoothly uniformly on compact subsets. Moreover, by the factϕ j−ϕ j+1≥ 1j− 1j+1 , it is clear that we may choose some sequence a j with ϕ j,a j ↓ϕ .From the above, it suffices now to prove the followingCLAIM: if ϕ ∈ C0(M)⋂C∞(U)⋂Psh(M,η) for some open set U and V is aprecompact open subset of U , there exists a sequence ϕ j ∈ C∞(M)⋂Psh(M,η)with ϕ j ↓ ϕ pointwise on M and smoothly uniformly on V .The claim follows from a slight modification of the proof of the main Theo-rem in [1]. In [1], an arbitrary open cover Uα of M is first chosen, then in eachUα a smooth local approximation ϕα,δ of ϕ is constructed through the use of localKa¨hler potentials and mollification. Then for fixed δ , a global smooth approxima-tion of ϕ on M is defined as the pointwise regularized maximum (from [15]) of theϕα,δ ’s where the maximum is taken over all α . Then, by letting δ → 0, it is shownthere exists a sequence ϕ j ∈C∞(M) with ϕ j ↓ ϕ pointwise on M. To have smoothconvergence uniformly on V we modify this construction slightly as follows.First we choose some finite open cover {Vα}kα=1 of the compact set M\U .Moreover, by compactness we may assume for each α > 0 we have a proper in-clusion of open sets Vα ⊂Uα ⊂Wα where: Wα⋂V = /0 and Wα is a holomorphiccoordinate neighborhood with smooth local Ka¨hler potential η = i∂ ∂¯ fα . Thenletting U0 :=U we take {Uα}kα=0 as our open cover of M.84Next we define local approximations ϕα,δ of ϕ on each Uα . For α = 0, let gαbe a smooth function which is equal to 0 in U\∪α 6=0 Vα and equals −1 outsidesome compact subset of U . For all α 6= 0, let gα be a smooth function in Uαsuch that gα = 0 in Vα and gα =−1 outside some compact subset of Uα . Assumethat i∂ ∂¯gα ≥ −Cω for some C independent of α . Now we define ϕα,δ on Uα asfollows. If α 6= 0, then as in [1] we letϕα,δ = uα,δ − fα +εCgαwhere uα,δ is a mollification of uα := ϕ+ fα ∈ Psh(Uα) in Wα so that uα,δ ↓ uα asδ → 0. If α = 0, then we letϕ0,δ = ϕ+εCg0In both cases, we have ϕα,δ ∈ Psh(Uα ,(1+ ε)η) and it is non-increasing as δdecreases.Now given our open cover {Uα}kα=0, and local approximations ϕα,δ in these,the proof of the claim follows exactly as in [1] involving the regularized maximum(from [15]) of the ϕα,δ ’s where the maximum is taken over all α . Then, by lettingδ → 0. We refer to [1] for details of this argument.85

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0367001/manifest

Comment

Related Items