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Geometric properties of the space of Lagrangian self-shrinking tori in ℝ⁴ Ma, Man Shun 2017

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Geometric properties of the space of Lagrangianself-shrinking tori in R4byMan Shun MaA 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 2017c©Man Shun Ma, 2017AbstractWe prove that any sequence {Fn : Σ→ R4} of conformally branched compact La-grangian self-shrinkers to the mean curvature flow with uniform area upper boundhas a convergent subsequence, if the conformal structures do not degenerate. WhenΣ has genus one, we can drop the assumption on non-degeneracy the conformalstructures. When Σ has genus zero, we show that there is no branched immer-sion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [52] indimension two by allowing branch points.When the area bound is small, we show that any such Lagrangian self-shrinkingtorus in R4 is embedded with uniform curvature estimates.For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori.Using the finiteness of entropy values, we construct a piecewise Lagrangianmean curvature flow for Lagrangian immersed tori, along which the Lagrangiancondition is preserved, area is decreasing, and the compact type I singularities witha fixed area upper bound can be perturbed away in finitely many steps. This is aLagrangian version of the construction for embedded surfaces in R3 in [17].In the noncompact situation, we derive a parabolic Omori-Yau maximum prin-ciple for a proper mean curvature flow when the ambient space has lower boundon `-sectional curvature. We apply this to show that the image of Gauss map ispreserved under a proper mean curvature flow in euclidean spaces with uniformbounded second fundamental form. This generalizes a result of Wang [53] forcompact immersions. We also prove a Omori-Yau maximum principle for properlyimmersed self-shrinkers, which improves a result in [8].iiLay SummaryIn this thesis we study the mean curvature flow of Lagrangian submanifolds.We show several compactness theorems for the space of compact Lagrangianself-shrinkers in R4. When we restrict to torus, we show that Lagrangian self-shrinking torus in R4 with small area is embedded with uniform curvature esti-mates.For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the self-shrinking tori. Using this, we constructa piecewise Lagrangian mean curvature flow for Lagrangian immersed tori.In the noncompact situation, we derive a parabolic Omori-Yau maximum prin-ciple for a proper mean curvature flow. We apply this to show that the image ofGauss map is preserved under a proper mean curvature flow in euclidean spaceswith uniform bounded second fundamental form.iiiPrefaceAll 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 3,4 are from [11], [12]. These are close and extensivecollaboration with Professor Jingyi Chen, with both authors contributing aboutequally. The first drafts of the manuscript [11], [12] were written by me, and thenrevised by both authors.Materials in Chapter 5 are from [38], where I am the sole author.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Compactness of the space of Lagrangian self-shrinking surfaces . 11.2 Finiteness of entropy and piecewise Lagrangian mean curvature flow 51.3 Parabolic Omori-Yau maximum principle and some applications . 82 Background in mean curvature flow and Lagrangian immersions . . 122.1 Mean curvature flow and self-shrinkers . . . . . . . . . . . . . . . 122.2 Lagrangian immersions . . . . . . . . . . . . . . . . . . . . . . . 132.3 F -stability and entropy stability . . . . . . . . . . . . . . . . . . 153 Compactness of the space of compact Lagrangian self-shrinking sur-faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Lagrangian branched conformal immersions . . . . . . . . . . . . 203.2 Proof of Theorem 1.1.2 . . . . . . . . . . . . . . . . . . . . . . . 253.3 Proof of Theorem 1.1.3 . . . . . . . . . . . . . . . . . . . . . . . 343.4 Lagrangian self-shrinking tori with small area . . . . . . . . . . . 36v4 Finiteness of entropy and piecewise Lagrangian mean curvature flow 404.1 A Łojasiewicz-Simon type gradient inequality for branched self-shrinking tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.1 A Fredholm operator of index zero . . . . . . . . . . . . . 414.1.2 A Łojasiewicz-Simon type inequality . . . . . . . . . . . 514.1.3 Proof of Theorem 1.2.1 . . . . . . . . . . . . . . . . . . . 564.2 Piecewise Lagrangian mean curvature flow . . . . . . . . . . . . . 574.2.1 Generalization to Lagrangian immersion of higher genussurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Proof of Lemma 4.1.5 . . . . . . . . . . . . . . . . . . . 654.3.2 Analyticity of E andM . . . . . . . . . . . . . . . . . . 665 Parabolic Omori-Yau maximum principle for mean curvature flow . 695.1 Proof of the parabolic Omori-Yau maximum principle . . . . . . . 695.2 Preservation of Gauss image . . . . . . . . . . . . . . . . . . . . 745.3 Omori-Yau maximum principle for self-shrinkers . . . . . . . . . 77Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80viAcknowledgmentsI would like to express my sincere gratitude to my advisor Professor Jingyi Chenfor the continuous support of my Ph.D study, for his patience, motivation, and im-mense knowledge. His guidance helped me in all the time of research and writingof this thesis.I would also like to thank my family: my parents and my wife Iris for support-ing me throughout writing this thesis and my life in general.viiChapter 1IntroductionThe thesis is divided into three parts. In the first part, we study the space of compactLagrangian self-shrinking surfaces inR4. We prove two compactness theorems andshow that Lagrangian self-shrinking tori with small area are embedded and haveuniform curvature estimates.In the second part, we study a dynamical property of Lagrangian self-shrinkingtori. We show that Lagrangian self-shrinking tori can attain only finitely many en-tropy values under an area upper bound. Then we define a piecewise Lagrangianmean curvature flow for Lagrangian immersed tori in R4 which preserves the La-grangian condition and the Maslov class, decreases area and avoids compact typeI singularities with any given area upper bound in finite steps.In the third part, we study mean curvature flow of noncompact immersions andderive a parabolic Omori-Yau maximum principle for proper mean curvature flow.1.1 Compactness of the space of Lagrangianself-shrinking surfacesOne of the major challenging problems in the study of Lagrangian mean curva-ture flow is to formulate a weak version of the mean curvature flow that preservesthe Lagrangian condition and goes beyond singular time, as the well-known weakforms of mean curvature flow such as the Brakke flow or the level set approach donot work well in the Lagrangian setting.1In general dimension and codimension, it is known that the rescaled flow ata finite time singularity converges weakly (up to subsequences) to a self-similarsolution of mean curvature flow {νt : t ∈ (−∞,0)} so that νt =√−tν−1. If ν−1 isa smooth immersion F , it is called a self-shrinker and it satisfies the self-shrinkingequation~H =−12F⊥, (1.1)where ~H is the mean curvature vector and F⊥ is the normal component of theposition vector F .Given a Lagrangian immersions F : Σn → R2n. It is shown in [51] that theLagrangian condition is preserved along the mean curvature flow when Σ is com-pact. Therefore, the self-shrinkers arise from Lagrangian mean curvature flow areLagrangian immersions. When n = 1, the Lagrangian condition is automaticallysatisfied by smooth curves and self shrinking solutions are studied in [1].In the first part of the thesis, we restrict our attention to compact Lagrangianself-shrinkers in R4. For arbitrary n, Smoczyk [52] showed that there is no La-grangian self-shrinking immersion in R2n with zero first Betti number. In partic-ular, there is no immersed Lagrangian self-shrinking sphere in R4. To establishcompactness properties of the moduli space of compact Lagrangian shrinkers, it iscrucial to generalize Smoczyk’s result to branched Lagrangian immersions as thelimit of a sequence of immersions may not be an immersion anymore.Indeed, the rigidity holds for branched Lagrangian self-shrinking spheres inR4:Theorem 1.1.1. There does not exist any branched conformal Lagrangian self-shrinking sphere in R4.At any immersed point, the mean curvature form αH = ι~Hω of a Lagrangianself-shrinker satisfies a pair of differential equations that form a first order ellipticsystem. The key ingredient in the proof of Theorem 1.1.1 is to show that αHextends smoothly across the branch points. The self-shrinker equation yields anL∞ bound on αH and this is useful to show that αH satisfies the first order ellipticsystem distributionally on S2. Smoothness of the extended mean curvature formthen follows from elliptic theory.2The main application of Theorem 1.1.1 is to derive compactness results of com-pact Lagrangian self-shrinkers.Let Σ be a fixed compact oriented smooth surface and Fn : Σ→ R4 be a se-quence of branched conformally immersed Lagrangian self-shrinkers in R4. Let〈·, ·〉 be the standard Euclidean metric on R4 and hn be the Riemannian metric onΣ which is conformal to the pull back metric F∗n 〈·, ·〉 on Σ such that either1. it has constant Gauss curvature −1 if the genus of Σ is greater than one, or2. (Σ,hn) is C/{1,a+bi} with the flat metric, where −12 < a≤ 12 , b≥ 0, a2+b2 ≥ 1 and a≥ 0 whenever a2+b2 = 1.It is well known that the moduli space of the conformal structures on Σ is parametrizedby metrics of the above form.We now state our compactness result.Theorem 1.1.2. Let Fn : (Σ,hn)→ R4 be a sequence of branched conformally im-mersed Lagrangian self-shrinkers with a uniform area upper bound Λ. Supposethat the sequence of metrics {hn} converges smoothly to a Riemannian metric hon Σ. Then a subsequence of {Fn} converges smoothly to a branched conformallyimmersed Lagrangian self-shrinker F∞ : (Σ,h)→ R4.Note that there is a universal positive lower bound on the extrinsic diametersfor two dimensional branched conformal compact shrinkers (cf. section 3.2). Thelimit F∞ cannot be a constant map.The proof of Theorem 1.1.2 uses the observation that self-shrinkers are in factminimal immersions into (R4,G), where G is a metric on R4 conformal to theEuclidean metric. The advantage of this viewpoint is that we are then able to usethe bubble tree convergence of harmonic maps developed in [42]. In particular,Theorem 1.1.1 shows that no bubble is formed during the process and thus theconvergence is smooth.It is interesting to compare Theorem 1.1.2 with the compactness results ofColding-Minicozzi [18] on embedded self-shrinkers in R3 and of Choi-Schoen[15], Fraser-Li [25] on embedded minimal surfaces in three dimensional manifoldN (with or without boundary) with nonnegative Ricci curvature. In their cases,3they use the local singular compactness theorem (see for example Proposition 2.1of [18]), which says that for any sequence of embedded minimal surfaces Σn in Nwith a uniform (global or local) area or genus upper bound, there is an embeddedminimal surface Σ such that a subsequence of {Σn} converges smoothly and locally(with finite multiplicities) to Σ away from finitely many points in Σ. A removablesingularity theorem, which is based on the maximum principle and is true onlyin the codimension one case, is needed to prove the local singular compactnesstheorem. Hence a similar statement is not available in higher codimension. Forembedded minimal surfaces in a 3-manifold, Colding and Minicozzi have provendeep compactness results ([16] and the reference therein).In the case of arbitrary codimension, the regularity of the limit of a sequence ofminimal surfaces (as a stationary varifold given by Geometric Measure Theory) isa subtle issue. Not much is known except for some special cases, such as Gromov’scompactness theorem on pseudo holomorphic curves [27].When Σ = T has genus one, we can drop the assumption on the convergenceof conformal structures in Theorem 1.1.2.Theorem 1.1.3. Let Fn :T→R4 be a sequence of branched conformally immersedLagrangian self-shrinking tori with a uniform area upper bound Λ. Then a sub-sequence of {Fn} converges smoothly to a branched conformally immersed La-grangian self-shrinking torus F∞ : T→ R4.Our strategy of proving Theorem 1.1.3 is to rule out the possibility of degen-eration of the conformal structures induced by the immersions Fn on T. To dothis, we use the general bubble tree convergence results in [10], [13] that allowthe conformal structures to degenerate. The key observation is that when the con-formal structures degenerate, some homotopically nontrivial closed curves in thetorus must be pinched to points. Thus the limiting surface is a finite union ofspheres (arising from collapse of the closed curves and from the bubbles at theenergy concentration points) which are branched Lagrangian self-shrinkers in R4,but Theorem 1.1.1 forbids their existence. Therefore, for the genus one case, theconformal structures cannot degenerate, and the desired result then follows fromTheorem 1.1.2.If Λ < 32pi , the Willmore functional of a self-shrinker with area upper bound4Λ is less than 8pi; a classical theorem of Li and Yau [36] then asserts that all suchLagrangian self-shrinking tori must be embedded and without branch points. Usingresults of Lamm-Scha¨tzle in [32] and Theorem 1.1.3, we show that the upper boundcan be increased above Li-Yau’s estimate. We introduce the following definition:Definition 1.1.1. Let Λ be a positive number. Let XΛ be the space of branchedconformally immersed Lagrangian self-shrinking tori with area less than or equalto Λ.Theorem 1.1.4. There are positive numbers ε0,ε1 and C0, where ε1 ≤ ε0, so that1. (No Branch Points) All elements in X32pi+ε0 are immersed, and all elementsin X32pi+ε1 are embedded.2. (Curvature Estimates) If F ∈ X32pi+ε0 , then the second fundamental form ofF is bounded by C0.1.2 Finiteness of entropy and piecewise Lagrangianmean curvature flowIn [17], Colding and Minicozzi introduce an entropy functional (see (2.13)) of ahypersurface (cf. [39]) and show that the sphere and the generalized cylinders arethe only entropy stable self-shrinking hypersurfaces. Using this and a compact-ness theorem [18] on the space of embedded self-shrinking surfaces in R3, theyconstructed in [17] a piecewise mean curvature flow for embedded surfaces in R3(under some assumptions), such that if a uniform diameter estimate holds then theflow shrinks to a round point.In [33], [34] the authors studied the Lagrangian entropy stability of Lagrangianself-shrinking immersions and obtained entropy instability results. In particular, Liand Zhang showed in [34] that if F : Mn→ R2n is a closed orientable Lagrangianself-shrinker and the first Betti number of M is greater than 1 then F is Lagrangianentropy unstable 1. Since there is no simply connected closed Lagrangian self-1More precisely, in [33], [34], the authors study the Lagrangian F -stability of a Lagrangianimmersion. ThatF -instability implies entropy instability is proved in [17] for the hypersurface caseand can be generalized to immersions of higher codimension. See [3] and also Chapter 2.5shrinker, all closed orientable Lagrangian self-shrinkers in R4 are Lagrangian en-tropy unstable.When the area upper boundΛ is not small (as in Theorem 1.1.4), it is not knownwhether any branched conformal Lagrangian self-shrinking torus with nonemptybranch locus exists or not. The possible existence of branch points of elements inXΛ is a serious obstacle for applications to Lagrangian mean curvature flow as onewould hope to perturb the branched Lagrangian surface to a nearby Lagrangianimmersion, but such resolution of singularity in the Lagrangian setting, even indimension two, is not available. Note that it is in general difficult to study nearbybranched immersions by deforming them along the normal vector fields. In par-ticular, it is hard to study stability problem of branched Lagrangian self-shrinkingimmersions as in [17], [34], and Weinstein’s Lagrangian neighbourhood theorem[54] does not apply to the branched case. In view of all these and the special featureof the embedded graphic representation of a surface near a self-shrinker in the codi-mension one case, the idea of the piecewise mean curvature flow introduced in [17]is not directly applicable to the Lagrangian case in R4, even with the compactnesstheorems 1.1.2, 1.1.3.In order to construct a piecewise Lagrangian mean curvature flow for a torus,we observe that one can bypass the issue of branchedness of a limiting surface inXΛ by controlling the entropy values λ (F) attained by the self-shrinkers, where fora branched immersion F : T→ R4 its entropy is defined byλ (F) = supx0∈R4,t0>014pit0∫Te−|F(x)−x0 |24t0 dµF .The theorem below is a crucial ingredient in our construction of piecewiseLagrangian mean curvarture flow for tori, but it is also interesting in its own right:it is equivalent to that in the induced metric from G = e−|x|24 δi j on R4 the areasof branched Lagrangian self-shrinking tori in XΛ can only take a finite number(depending on Λ) of values for any given Λ.Theorem 1.2.1. Let λ : XΛ→ [0,∞) be the entropy function which sends F to itsentropy λ (F). Then the image of λ is finite for any given Λ.To prove Theorem 1.2.1, we derive a Łojasiewicz-Simon gradient inequality for6branched conformal self-shrinking 2-dimensional tori. The celebrated Łojasiewicz-Simon gradient inequality is proved in [49] with important applications to theharmonic map flow and the minimal cones. Since the pioneering work [49], theinequality and its variations have wide applications in geometric problems. Formean curvature flow, Schulze [48] used the inequality to prove a uniqueness resultfor compact embedded singularity of tangent flow. Colding and Minicozzi [19]derived a Łojasiewicz gradient inequality in a noncompact setting and settled theuniqueness problem for all generic singularities of mean convex mean curvatureflow at all singularities.The classical Łojasiewicz-Simon gradient inequality is established for real an-alytic functionals over a compact manifold whose Euler-Lagrange operator is el-liptic and of order 2. In our case, we are concerned with the entropy functional λ ,which is, at a self-shrinker, just the area of the shrinker in (R4,G) up to a universalconstant. However, in our situation, the self-shrinkers might be branched and theEuler-Lagrange operator of the area functional fails to be elliptic at the branch lo-cus, so Simon’s infinite dimensional version of the Łojasiewicz inequality in [49]is not directly applicable. To overcome the difficulty, we consider the real analyticenergy functional E defined on the mapping space C2,α(T,R4) together with theTeichmu¨ller space of T, and continue to view self-shrinkers as branched minimalimmersions in (R4,G). The functional E has been extensively used in minimal sur-face theory, especially, in showing existence of minimal surfaces. A critical pointof E corresponds to a branched conformal self-shrinking torus. Since the space ofconformal structures on a torus is two dimensional, the ellipticity of the L2-gradientof E at a critical point of E for each fixed conformal structure enables us to showthat the second order derivativeL of E at the critical point is a Fredholm operatorof index zero, which is sufficient to derive the desired gradient inequality. Theorem1.2.1 is then a direct consequence of the gradient inequality and the compactnessTheorem 1.1.3.We then apply Theorem 1.2.1 to construct a piecewise Lagrangian mean cur-vature flow for Lagrangian immersed tori F : T→ R4 (see Definition 4.2.1). Weshow that all type I singularities with an arbitrarily given area upper bound can beperturbed in finitely many steps, where a smooth Lagrangian mean curvature flowof a torus restarts at each step, such that the same kind of singularities will not7appear in the last step. We remark that the perturbation can be made arbitrarilysmall while fixing the number of perturbations performed. Note that, in the specialcase of small area, Theorem 1.1.4 is sufficient since the existence of a nearby La-grangian immersion of the torus around a limiting surface in XΛ (now immersed)follows from the Lagrangian neighbourhood theorem.Our main result on Lagrangian mean curvature flow in a weak form isTheorem 1.2.2. Let F :T→R4 be an immersed Lagrangian torus and let Λ,δ > 0be given constants. Then there exists a piecewise Lagrangian mean curvature flow{F it : i = 0,1, · · · ,k− 1} with initial condition F, where k ≤ |λ (XΛ)| < ∞, suchthat the singularity at time tk is not a type I singularity modelled by a compactself-shrinker with area less than or equal to Λ. Moreover, the Maslov class of eachimmersion is invariant along the flow.Under an additional assumption, we prove a similar result in Theorem 4.2.1 forthe case of genus larger than one.1.3 Parabolic Omori-Yau maximum principle and someapplicationsIn the last chapter, we consider mean curvature flow of noncompact manifold. Themain result is a parabolic Omori-Yau maximum principle for mean curvature flowof noncompact manifold. First we recall the Omori-Yau maximum principle forthe Laplace operator.Let (M,g) be a Riemannian manifold and let u : M→ R be a twice differen-tiable function. If M is compact, u is maximized at some point x∈M. At this point,basic advanced calculus impliesu(x) = supu, ∇Mu(x) = 0, ∆Mu(x)≤ 0.Here ∇M and ∆M are respectively the gradient and Laplace operator with respect tothe metric g. When M is noncompact, a bounded function might not attain a max-imum. In this situation, Omori [41] and later Yau [56] provide some noncompactversions of maximum principle. We recall the statement in [56]:8Theorem 1.3.1. Let (M,g) be a complete noncompact Riemannian manifold withbounded below Ricci curvature. Let u : M→R be a bounded above twice differen-tiable function. Then there is a sequence {xi} in M such thatu(xi)→ supu, |∇u|(xi)→ 0, limsupi→∞∆Mu(xi)≤ 0.Maximum principles of this form are called Omori-Yau maximum principles.The assumption on the lower bound on Ricci curvature in Theorem 1.3.1 has beenweaken in, e.g., [9], [43]. On the other hand, various Omori-Yau type maximumprinciples have been proved for other elliptic operators and on solitons in geometricflows, such as Ricci solition [8] and self-shrinkers in mean curvature flows [14].The Omori-Yau maximum principles are powerful tools in studying noncompactmanifolds and have a lot of geometric applications. We refer the reader to the book[2] and the reference therein for more information.In this paper, we derive the following parabolic version of Omori-Yau maxi-mum principle for mean curvature flow.Theorem 1.3.2 (Parabolic Omori-Yau Maximum Principle). Let n≥ 2 and m≥ 1.Let (Mn+m, g¯) be an n+m-dimensional noncompact complete Riemannian mani-fold such that the (n− 1)-sectional curvature of M is bounded below by −C forsome positive constant C. Let Mn be a n-dimensional noncompact manifold and letF : Mn× [0,T ]→M be a proper mean curvature flow. Let u : M× [0,T ]→ R be acontinuous function which satisfies1. sup(x,t)∈M×[0,T ] u > supx∈M u(·,0),2. u is twice differentiable in M× (0,T ], and3. (sublinear growth condition) There are B > 0, α ∈ [0,1) and some y0 ∈ Mso thatu(x, t)≤ B(1+dM(y0,F(x, t))α), ∀(x, t) ∈M× [0,T ]. (1.2)9Then there is a sequence of points (xi, ti) ∈M× (0,T ] so thatu(xi, ti)→ supu, |∇Mti u(xi, ti)| → 0, liminfi→∞(∂∂ t−∆Mti)u(xi, ti)≥ 0. (1.3)We remark that the above theorem makes no assumption on the curvature ofthe immersion Ft . See section 5.1 for the definition of `-sectional curvature.With this parabolic Omori-Yau maximum principle, we derive the followingresults.In [53], the author studies the gauss map along the mean curvature flow in theeuclidean space. He shows that if the image of the gauss map stays inside a totallygeodesic submanifold in the Grassmanians, the same is also true along the flowwhen the initial immersion is compact. As a first application, we extend Wang’stheorem to the noncompact situation.Theorem 1.3.3. Let F0 : Mn → Rn+m be a proper immersion and let F : Mn×[0,T ]→ Rn+m be a mean curvature flow of F0 with uniformly bounded secondfundamental form. Let Σ be a compact totally geodesic submanifold of the Grass-manians of n-planes in Rn+m. If the image of the Gauss map γ satisfies γ(·,0)⊂ Σ,then γ(·, t)⊂ Σ for all t ∈ [0,T ].As a corollary, we have the following:Corollary 1.3.1. Let F0 : Mn → R2n be a proper Lagrangian immersion and letF : M× [0,T ]→ R2n be a mean curvature flow with uniformly bounded secondfundamental form. Then Ft is Lagrangian for all t ∈ [0,T ].The above result is well-known when M is compact [51], [53]. Various formsof Corollary 1.3.1 are known to the experts (see remark 5 in Chapter 5).The second application is to derive a Omori-Yau maximum principle for theL -operator of a proper self-shrinker. The L operator is introduced in [17] whenthe authors study the entropy stability of a self-shrinker. Since then it proves to bean important operator in mean curvature flow. Using Theorem 1.3.2, we proveTheorem 1.3.4. Let F˜ : Mn→ Rn+m be a properly immersed self-shrinker and letf : Mn→ R be a twice differentiable function so thatf (x)≤C(1+ |F˜(x)|α) (1.4)10for some C > 0 and α ∈ [0,1). Then there exists a sequence {xi} in M so thatf (xi)→ supMf , |∇ f |(xi)→ 0, limsupi→∞L f (xi)≤ 0. (1.5)The above theorem is a generalization of Theorem 5 in [8] since we assumeweaker conditions on f .11Chapter 2Background in mean curvatureflow and Lagrangian immersions2.1 Mean curvature flow and self-shrinkersA family of immersions Ft : Σ→ RN from an n-dimensional manifold Σ to theEuclidean space is said to satisfy the mean curvature flow if∂Ft∂ t= ~H. (2.1)Here ~H is the mean curvature vector given by ~H = trA, the trace of the secondfundamental formA(X ,Y ) = (DXY )⊥(here D is the standard connection on Rn and ⊥ denotes the normal component ofa vector with respect to the immersion Ft). In local coordinates (x1. · · ·xn) of Σ, thesecond fundamental form A and the mean curvature vector ~H are given byAi j =(∂i jF)⊥, ~H =n∑i, j=1gi jAi j = ∆gFwhere ∆g is the Laplace-Beltrami operator in the induced metric g.When M is compact, standard parabolic PDE theory implies that the mean12curvature flow exists for a short time [0,T ) and is unique.An immersion is called self-shrinking (or a self-shrinker) if it satisfies~H =−12F⊥. (2.2)If F is self-shrinking, then up to a family of diffeomorphisms, the family of im-mersions{√−tF : t ∈ [−1,0)}solves the mean curvature flow. The self-shrinkers model the singularity of meancurvature flow (cf. [29, 30, 55]).Let G be a metric on Rn+k defined byG(·, ·) = e− |x|22n 〈·, ·〉. (2.3)The following lemma is well-known and is proved by Angenent [4] for hyper-surfaces. The proof can be generalized to arbitrary codimension.Lemma 2.1.1. The immersion F satisfies equation (2.2) if and only if F is a mini-mal immersion with respect to the metric G defined in (2.3) on Rn+k.2.2 Lagrangian immersionsNext we consider immersions F : Σ→ R2n. F is called Lagrangian if F∗ω = 0,whereω =n∑i=1dxi∧dyiis the standard symplectic form onR2n. Let 〈·, ·〉,J be the standard euclidean metricand complex structure on R2n respectively. It is known that ω , J and 〈·, ·〉 arerelated by〈JX ,JY 〉= 〈X ,Y 〉, ω(X ,Y ) = 〈JX ,Y 〉 (2.4)for any X ,Y ∈ TxR2n,x ∈ R2n. Thus F is Lagrangian if and only if J sends thetangent vectors of Σ to the normal vectors. In particular, by (2.4), J~H(x) is tangentto F(Σ) at x for any Lagrangian immersion F . The mean curvature form αH is the131-form on Σ defined by: for all x ∈ Σ and Y ∈ TxΣ,αH(Y ) = ω(~H(x),(F∗)xY ) = g(J~H(x),(F∗)xY ). (2.5)Let d denotes the exterior differentiation of Σ. For Lagrangian immersions, it isshown in [20] that αH is closed:dαH = 0 . (2.6)If F is also a self-shrinker, we have the following equation for αH :Lemma 2.2.1. Let F be a Lagrangian self-shrinker. Then αH satisfiesd∗αH =−14αH(∇g|F |2), (2.7)where d∗ is the formal adjoint of d on Σ with respect to g.Proof. (see also [7], p.1521) In local coordinates (x1, · · · ,xn) of Σ,αH =n∑i=1(αH)idxiwhere the coefficients are given by(αH)i = 〈J~H,∂iF〉= −12〈JF⊥,∂iF〉= −12〈JF,∂iF〉.Note that we have used (2.2) in the second equality. Now fix a point p∈ Σ and take14the normal coordinates at p. At p, gi j = gi j = δi j and Γki j = 0. At p, we haved∗αH =−m∑i, j=1gi j(αH)i; j=−m∑i=1(αH)i,i=m∑i=112∂i〈JF,∂iF〉=m∑i=112(〈J∂iF,∂iF〉+ 〈JF,∂ 2ii F〉) .Using J∂iF ⊥ ∂iF and ∂ 2ii F = Aii for each i at p,d∗αH =12m∑i=1〈JF,Aii〉=12〈JF, ~H〉=−12〈F,J~H〉=−12αH(F>)=−14αH(∇g|F |2),where F> is the tangential component of F and we have usedF> =m∑i=1〈F,∂iF〉∂iF = 12∇g|F |2at p.2.3 F -stability and entropy stabilityThe entropy λ andF -stability are introduced in [17] for an embedded self-shrinkinghypersurfaces and are later carried over in [3], [33], [34] for all codimensions.The Lagrangian case is discussed in [33], [34] and the definition of LagrangianF -stability is introduced therein. We start with recalling the definition of the F -15functional, the entropy λ and the related stability. When we consider Lagrangianimmersions, we will assume k = n.Definition 2.3.1. Let (x0, t0) ∈ Rn+k×R>0. The F -functional of an immersionF : Σn→ Rn+k is given byFx0,t0(F) = (4pit0)− n2∫Σe−|F(x)−x0 |24t0 dµF . (2.8)TheF -functional characterizes the self-shrinkers as follows: F : Σ→ Rn+k isa self-shrinker if and only ifddsFxs,ts(Fs)∣∣∣∣s=0= 0for all variations (xs, ts,Fs) such that (x0, t0,F0) = (0,1,F).We recall that a normal vector field X along a Lagrangian immersion is calleda Lagrangian variation ifd(ιXω) = 0. (2.9)Definition 2.3.2. A self-shrinker F is called (Lagrangian) F -stable if for all (La-grangian) variations Fs, there is a variation (xs, ts) of (0,1) so thatd2ds2Fxs,ts(Fs)∣∣∣∣s=0≥ 0.In [34], Li and Zhang calculate the second variation of the F -functional of aLagrangian immersion with respect to the Lagrangian variations. They prove:Theorem 2.3.1. Let Σ be a compact orientable n-dimensional manifold whose firstBetti number is greater than 1. If F : Σ→ R2n is a Lagrangian self-shrinker, thenF is LagrangianF -unstable.When F : Σ→ R2n is a Lagrangian immersion, let Fs : Σ→ R2n be a normalvariation of F such that each Fs is a Lagrangian immersion. In this case, the normalvariational vector field X = dds |s=0Fs can be identified with a closed 1-form on Σ byX 7→ −ιXω . The converse is also true as seen in the following elementary lemma.16Recall that if α is a 1-form on a Riemannian manifold (Σ,g) then α] is thevector field on Σ uniquely determined byg(α],Y ) = α(Y ), ∀Y ∈ TΣ. (2.10)Lemma 2.3.1. Let F : Σn→R2n be a compact Lagrangian immersion and let α bea closed 1-form on Σ. Then there is a family of Lagrangian immersions Fs :Σ→R2nso that F0 = F anddds∣∣∣∣s=0Fs = Jα]. (2.11)Proof. Let pi : NΣ→ Σ be the normal bundle of the immersion F . Then the map-pingF˜(x,v) = F(x)+ vis a local diffeomorphism from a tubular neighbourhood U of the zero section ofNΣ onto its image in R2n.Since α is a closed 1-form on Σ, β = (pi|U)∗α is a closed 1-form on U , and βsends the normal vectors v to zero. The pullback 2-form ω0 = F˜∗ω on U is closedas ω is closed and it is non-degenerate as F˜ is a locally diffeomorphic and ω isnon-degenerate. Let X be the vector field on U dual to β with respect to ω0, thatis,β (Y ) =−ω0(X ,Y ) (2.12)for all vector fields Y on U . Let φs with s ∈ (−ε,ε) be the one parameter group ofdiffeomorphisms on U generated by X . Then Fs := F˜ ◦φs|Σ : Σ→ R2n is a familyof Lagrangian immersions in R2n and F0 = F˜ ◦φ0|Σ = F˜ |Σ = F .It remains to verify (2.11). By the definition of F˜ , its differential F˜∗ mapsthe tangent vectors to the zero section (Σ,0) at the point (x,0) ∈U to the tangentvectors to the image surface F(Σ) at the point F(x) ∈ R2n and it maps the normalvectors to the normal vectors by the identity map at the corresponding points. Weneed to check X = Jα]. Let Y1,Y2 be arbitrary tangent vectors to the zero section(Σ,0) at a point (x,0). Since JY2 is normal to Σ as Σ is Lagrangian andω(X ,Y ) = 〈JX ,Y 〉,17we haveα(pi∗Y1) = β (Y1+ JY2)=−ω0(X ,Y1+ JY2)=−ω(F˜∗X , F˜∗Y1+ F˜∗JY2)=−ω(F˜∗X , F˜∗Y1+ JY2)=−〈JF˜∗X , F˜∗Y1+ JY2〉=−〈JF˜∗X , F˜∗Y1〉−〈F˜∗X ,Y2〉As F˜ is locally diffeomorphic, X is normal to the zero section because Y2 is arbi-trary. Then it follows from the arbitrariness of Y1 that −JX = α], by dropping thenotion F˜∗. This is the same as X = Jα].The entropy of a hypersurface is defined in [17, 39]. The definition for animmersion in any codimension is the same.Definition 2.3.3. The entropy of an immersion F : Σ→ Rn+k is defined asλ (F) = supx0,t0Fx0,t0(F). (2.13)It is clear that λ (F) is invariant under translations and scalings. Huisken’smonotonicity formula [29] implies that λ (Ft) is non-increasing if {Ft} satisfies themean curvature flow, and is constant if and only if {Ft} is self-shrinking. Anal-ogous to the entropy stability introduced in [17], we define Lagrangian entropystability of a Lagrangian self-shrinker.Definition 2.3.4. Let F : Σ→ R2n be a self-shrinker. Then F is called Lagrangianentropy stable if λ (F˜)≥ λ (F) for all Lagrangian immersions C0 close to F .In [17], it is proved that every F -unstable embedded self-shrinking hypersur-face which does not split off a line is entropy unstable. As observed in [3], the ex-act same proof works for any codimension. According to [34], the second variationformula for theF -functional at a closed Lagrangian self-shrinker can be rewrittenin terms of the closed 1-form dual to the Lagrangian variation field. Therefore,18when F : Σ→ R2n is a Lagrangian F -unstable self-shrinker, there is a closed 1-form α on Σ so thatF ′′(α)< 0 for all variations (xs, ts) of (0,1). To proceed fromthe Lagrangian F -instability to the Lagrangian entropy instability, one needs touse the actual family Fs of Lagrangian immersions coming from the Lagrangianvariation. By Lemma 2.3.1, there is a Lagrangian variation {Fs} that correspondsto α . By taking a family of diffeomorphism φs : Σ→ Σ, we can further assume that{Fs} is a family of normal variations. Thus the same proof of Theorem 0.15 in [17]can be carried over to show that F : Σ→ R2n is also Lagrangian entropy unstable.We omit the proof here.Theorem 2.3.2. Let Σ be compact and F : Σ→ R2n be an immersed Lagrangianself-shrinker. If F is Lagrangian F -unstable, then it is also Lagrangian entropyunstable. In particular, there is a Lagrangian immersion F̂ : Σ → R2n so thatλ (F) > λ (F̂). Moreover, F̂ can be chosen to be arbitrarily close to F, in thesense of smallness of ‖F− F̂‖Ck for any k.19Chapter 3Compactness of the space ofcompact Lagrangianself-shrinking surfaces3.1 Lagrangian branched conformal immersionsFrom now on we consider n = 2. Let (Σ,g0) be a smooth Riemann surface. Asmooth map F : Σ→ R4 is called a branched conformal immersion if1. there is a discrete set B⊂ Σ such that F : Σ\B→ R4 is an immersion,2. there is a function λ : Σ→ [0,∞) such that g := F∗〈 , 〉 = λg0 on Σ, whereλ is zero precisely at B, and3. the second fundamental form A on Σ\B satisfies |A|g ∈ L2(K \B,dµ) for anycompact domain K in Σ, where | · |g and dµ are respectively the norm andthe area element with respect to g (Note that g defines a Riemannian metricon Σ\B but not on Σ).Elements in B are called the branch points of F . A branched conformal im-mersion F : Σ→ R4 is called Lagrangian and self-shrinking if it is Lagrangianand self-shrinking, respectively, when restricted to Σ\B. Note that when F is La-grangian, the mean curvature form αH is defined only on Σ\B.20The following proposition is the key result on removable singularity of αH .Note that in this proposition we do not assume Σ to be compact.Proposition 3.1.1. Let F : Σ → R4 be a branched conformal Lagrangian self-shrinker with the set of branch points B. Then there is a smooth one form α˜ onΣ which extends αH and dα˜ = 0 on Σ.Proof. The result is local, so it suffices to consider Σ to be the unit disc D witha branch point at the origin only. Let (x,y) be the local coordinate of D. Wewrite αH = adx+ bdy for some smooth functions a and b on the punctured diskD∗ = {z ∈ D : z 6= 0}. Letdiv(αH) =∂a∂x+∂b∂y, ∇0 =(∂∂x,∂∂y)(3.1)be the divergence and the gradient with respect to the Euclidean metric δi j on D.As F is conformal, gi j = λδi j, where λ = 12 |DF |2. By restricting to a smaller diskif necessary, we assume that the image |F | and λ are bounded. By (2.2) we have|~H| ≤ 12|F |,where | · | is taken with respect to 〈·, ·〉 on R4. As g = F∗〈·, ·〉, using (2.5) we seethat|αH |g ≤C on D∗.Using gi j = λδi j and n = 2, we have∇g = λ−1∇0 and d∗g =−λ−1div.Hence equation (2.7) is equivalent todiv(αH) =14αH(∇0|F |2). (3.2)Moreover, as |αH |g =√λ−1(a2+b2), we also have|a|, |b| ≤ |αH |g√λ ≤C√λ .21Thus |a|, |b| are bounded on D∗. To simplify notations, letP =14αH(∇0|F |2).Note that P is also bounded on D∗.Both equations (2.6) and (3.2) are satisfied pointwisely in D∗. We now showthat they are satisfied in the sense of distribution on D. That is, for all test functionsφ ∈C∞0 (D), ∫DαH ∧dφ = 0 (3.3)and ∫DαH(∇0φ)dxdy =−∫DPφdxdy. (3.4)Note that all the integrands in equation (3.3) and (3.4) are integrable, since a,b andP are in L∞(D).First we show (3.3). Let ψr ∈C∞0 (D), r < 1/2, be a cutoff function such that0≤ ψ ≤ 1, |∇0ψ| ≤ 2/r andψ(x) =1 when |x| ≥ 2r,0 when |x| ≤ r.Then φψr ∈C∞0 (D∗)∩C∞(D). Using Stokes’ theorem and equation (2.6), we have0 =∫D∗d(φψrαH) =∫D∗d(φψr)∧αH .This implies ∫D∗ψrαH ∧dφ =−∫D∗φαH ∧dψr. (3.5)Since ψr→ 1 on D∗ as r→ 0 and αH , dφ are bounded,limr→0∫D∗ψrαH ∧dφ =∫D∗αH ∧dφ =∫DαH ∧dφby Lebesgue’s dominated convergence theorem. To estimate the right hand side of(3.5), note that as dψr has support on D2r \Dr, where Ds denotes the disk of radius22s. Hence ∣∣∣∣∫D φαH ∧dψr∣∣∣∣≤ 4C sup |φ |r∫D2r\Drdx = 12piC sup |φ |r→ 0 (3.6)as r→ 0. Thus (3.3) holds.To show (3.4), we use the same cutoff function ψr. Then φψr ∈ C∞0 (D∗)∩C∞(D). By the divergence theorem,0 =∫D∗div(φψrαH)dxdy =∫D∗αH(∇0(φψr))dxdy+∫D∗φψrdiv(αH)dxdy.Now we use (3.2) to conclude−∫D∗Pφψr dxdy =∫D∗ψrαH(∇0φ)dxdy+∫D∗φαH(∇0ψr)dxdy. (3.7)Similarly, we can estimate the second term on the right hand side of (3.7) as for(3.6): ∣∣∣∣∫D∗ φαH(∇0ψr)dxdy∣∣∣∣≤ 12piC sup |φ |r. (3.8)Using Lebesgue’s dominated convergence theorem again, we can set r→ 0 in (3.7)to arrive at (3.4).Writing αH = adx+bdy, the two equations (3.3) and (3.4) are equivalent to∫D(a∂φ∂y−b∂φ∂x)dxdy = 0, (3.9)∫D(a∂φ∂x+b∂φ∂y)dxdy =−∫DPφ dxdy, (3.10)for any test functions φ ∈C∞0 (D).For any ψ ∈C∞0 (D), set φ = ∂ψ∂y in (3.9), φ = ∂ψ∂x in (3.10) and cancel the crossterm b ∂2ψ∂x∂y by taking summation of the two, we have∫Da∆ψ dxdy =−∫DP∂ψ∂xdxdywhere ∆ is the Laplace operator in the Euclidean metric on D.Similarly, set φ = ∂ψ∂x in (3.9), φ =∂ψ∂y in (3.10) and take the difference of the23two equations, we obtain∫Db∆ψ dxdy =−∫DP∂ψ∂ydxdy.We conclude now that a and b satisfy∆a =∂P∂x,∆b =∂P∂y(3.11)on D in the sense of distribution. Now we apply the elliptic regularity theory fordistributional solutions. As F is smooth and α ∈ L∞(D), we have P∈ L2(D). Hencethe right hand side of equation (3.11) is in H loc−1 (D). By the local regularity theorem([24], Theorem 6.30), we have a,b ∈ H loc1 (D). This implies P ∈ H loc1 (D). Usingthis, we see that the right hand side of equation (3.11) is in H loc0 (D). By the samelocal regularity theorem again, these implies a,b ∈ H loc2 (D). Thus we can iteratethis argument and see that a,b∈H locs (D) for all positive integers s. By the Sobolevembedding theorem we have a,b ∈C∞(D). Hence αH can be extended to a smoothone form α˜ on D and dα˜ = 0 is satisfied on D.Now we proceed to the proof of Theorem 1.1.1.Proof. Let F : S2 → R4 be a branched conformal Lagrangian self-shrinker withbranch points b1, · · · ,bk. By Proposition 3.1.1, there is a smooth 1-form α˜ on S2such that α˜ = αH on S2 \B. As α˜ is closed and the first cohomology group of S2is trivial, there is a smooth function f on S2 such that d f = α˜ . By equation (2.7),f satisfies∆g f =−14 d f (∇g|F |2) (3.12)on S2 \B. Note that this equation is elliptic but not uniformly elliptic on S2 \B.By the strong maximum principle, the maximum of f cannot be attained in S2 \Bunless f is constant. Let b ∈ B be a point where f attains its maximum. Let D be alocal chart around b such that gi j = λδi j on D∗. As ∆g = λ−1∆ and ∇g = λ−1∇0,24equation (3.12) can be written∆ f =14d f (∇0|F |2) on D∗. (3.13)As f and |F |2 are smooth on D, the equation (3.13) is in fact satisfied on D. By thestrong maximum principle, f is constant as f has an interior maximum at b. HenceαH = 0 and ~H = 0. This implies that F is a branched minimal immersion in R4,which is not possible as S2 is compact.3.2 Proof of Theorem 1.1.2Let {Fn : (Σ,hn)→ R4} be a sequence of branched conformal Lagrangian self-shrinkers which satisfy the hypothesis in Theorem 1.1.2. In the following proof,we will view each self-shrinker Fn as a harmonic map from (Σ,hn) to (R4,G). Theexistence of harmonic 2-spheres in [44] and the bubble tree convergence theoremin [42] require compact target space. To deal with non-compactness of (R4,G), wewill show that a uniform area bound of the sequence {Fn} implies that all Fn(Σ) liein a bounded region. Hence the harmonic maps can be viewed as mappings into acompact Riemannian manifold. This is done in the next two lemmas.Lemma 3.2.1. Let F be a compact branched conformal self-shrinker in R4. Thenthe image of F lies in a ball of radius R0 centered at the origin in R4, where R0depends only on µ(F), the area of F.Proof. Let F : Σ→ R4 be a branched conformally immersed self-shrinker. By theself-shrinking equation (2.2),∆g|F |2 =−|F⊥|2+4 (3.14)holds on Σ\B, where g = F∗〈·, ·〉 and B is the finite branch locus.First, we show that F must intersect the closed ball centered at the origin of R4with radius 2. Since F is a branched conformal immersion, there is a nonnegativesmooth function ϕ and a smooth metric g0 on Σ compatible with the conformal25structure h so that g = ϕg0. Thereforeϕ ∆g = ∆g0and by (3.14),∆g0 |F |2 = ϕ (−|F⊥|2+4). (3.15)Unlike (3.14), (3.15) is satisfied everywhere on Σ, as both sides of the equationare continuous and B is finite. Since Σ is compact, the smooth function |F |2 attainsits minimum, say at x0 ∈ Σ. Since F is a minimal immersion in (R4,G), the tan-gential component F> is well defined even at a branch point and F>(x0) = 0. If Fis immersed at x0, the weak maximum principle shows that ∆g|F |2(x0) ≤ 0. Thisimplies |F⊥(x0)|2 ≤ 4 by (3.14). Since F>(x0) = 0,|F(x0)|2 = |F⊥(x0)|2 ≤ 4and we are done. Hence we only need to rule out the case that F is branched at x0,|F(x0)|2 > 4 and there does not exist any immersed point y ∈ Σ so that |F(y)|2 =|F(x0)|2. Assume this case happens. Since the branch points are isolated, |F |2 hasa strict minimum at x0. Noting again F>(x0) = 0 and |F |2 = |F>|2 + |F⊥|2, wehave |F⊥(x)|2 > 4 in a neighbourhood of x0. By (3.15) we have ∆g0 |F |2 ≤ 0 in theneighbourhood. However, this contradicts the strong maximum principle and weare done.Next, we show that the extrinsic distance between any two points on the imageof F is bounded above by a constant that depends only on the area upper bound.Note that∆g|F |2dµg = d ∗g d|F |2. (3.16)and the Hodge star operator ∗g depends only on the conformal class of g, ∆g|F |2dµgis well-defined on Σ. Thus we integrate (3.14) and use (2.2) to getW (F) :=14∫Σ|~H|2dµ = 14µ(F). (3.17)One also note that Simon’s diameter estimate [50] holds for 2-varifolds with squareintegrable generalized mean curvature ((A.16) in [31]). Thus there is a constant C26such that (µ(F)W (F)) 12≤ diamF(Σ)≤C(µ(F)W (F)) 12 , (3.18)wherediamF(Σ) := supx,y∈Σ|F(x)−F(y)|.Together with (3.17), we see thatdiamF(Σ)≤ 12Cµ(F).It follows that the image of F lies in B(R0) for some R0 depending only on thearea upper bound.Let U = BR0+1 endowed with the metric G given byG = e−|x|24 〈·, ·〉. (3.19)The next lemma enables us to apply the results in [42] for harmonic maps intoa compact Riemannian manifold.Lemma 3.2.2. There is a compact Riemannian manifold (N,g) such that (U,G)defined as above can be isometrically embedded into (N,g).Proof. Let d = 1R0+1 and N is the disjoint union of BR0+2 and Bd , with the identifi-cation that x∼ y if and only if y= x|x|2 by the inversion. The manifold N is compact,as it can be identified as the one point compactification of R4 via the stereographicprojection. Let g1 be any metric on Bd . Let ρ1,ρ2 ∈C∞(N) be a partition of unitysubordinate to the open cover {BR0+2,Bd} in N and define a Riemannian metric onN byg = ρ1G+ρ2g1.AsBR0+2∩Bd = {x ∈ BR0+2 : R0+1 < |x|< R0+2},g¯ = ρ1G+ρ2g1 = G on BR0+1 ⊂ BR0+2. Thus the inclusion U ⊂ BR0+1 ⊂ N is anisometric embedding of U .27The use of harmonic map theory is essential in our proof of Theorem 1.1.2.For the reader’s convenience, we recall the terminologies in the construction of thebubbles and the bubble tree for a sequence of harmonic maps from surfaces. Themain references are [44], [42].A smooth map f : (Σ,h)→ (N, g¯) from a two dimensional Riemannian surfaceto a Riemannian manifold is called harmonic if it is a critical point of the energyfunctionalEh,g¯( f ) =12∫Σ‖d fx‖2dµh,where d fx : TxΣ→ Tf (x)N is the differential of f and ‖d fx‖2 is locally given by‖d fx‖2 = gαβ ( f (x))hi j(x)∂ fα∂xi(x)∂ f β∂x j(x).It is well-known [47]:Proposition 3.2.1. Let f : (Σ,h)→ (N, g¯) be a mapping from a two dimensionalRiemannian surface to a Riemannian manifold. Then1. the energy is conformally invariant: Eh,g¯( f ) = Eeνh,g¯( f ).2. If f is a branched conformal immersion, then Eh( f ) = µ( f ), where µ( f ) isthe area of f .3. If f is conformal, then f is a harmonic map if and only if it is a branchedminimal immersion.4. If f is a nontrivial harmonic map from a 2-sphere, then f is conformal.As in [42], we state the analytic results in [44] that are needed in our proof ofTheorem 1.1.2 and in the bubble tree convergence [42].Proposition 3.2.2. There are positive constants C and ε0 that depend on (Σ,h) and(N, g¯) such that1. (Sup Estimate) If f : Σ→ N is harmonic with ∫D(2r) ‖d f‖2dµh < ε0, thensupD(r)‖d fx‖2 ≤Cr−2∫D(2r)‖d f‖2dµh. (3.20)282. (Uniform Convergence) If { fn} is a sequence of harmonic maps from a diskD(2r) with energies less than ε0 for all n, then a subsequence of { fn} con-verges in C1 in D(r).3. (Energy Gap) Any nontrivial harmonic map f : S2→ N has energy E( f ) ≥ε0.4. (Removable Singularities) Any smooth harmonic map from a punctured diskD∗ with finite energy extends to a smooth harmonic map on D.Let fn : (Σ,hn)→ (N, g¯) be a sequence of harmonic maps with a uniform energyupper bound E0. Assume that the metrics hn converge smoothly to a smooth metrich on Σ. The set of bubbling points (or energy concentration points) of { fn} isdefined asS = S{ fn} =⋂δ>0{x ∈ Σ : liminfn→∞∫D(x,δ )‖d fn‖2dµhn ≥ ε0}, (3.21)where ε0 is the constant in (1) of Proposition 3.2.2. Since hn→ h, Theorem 2.3 in[45] asserts that { fn}, taking a subsequence if necessary, converges to a harmonicmap f in C1(M−S,N) and S consists of finitely many points.Next we focus on each x ∈ S and describe how the bubble tree is constructedat x. In [42], a sequence εn→ 0 is chosen and via the exponential map at x, all thegeodesic disks D(0,2εn) are identified as subsets in R2 ∼= TxΣ. Let cn be the centerof mass of ‖d fn‖2dhn and λn > 0 be suitably chosen (see Section 1 in [42] for thechoices of cn,λn) so thatD(cn,nλn)⊂ D(cn,εn)⊂ D(0,2εn).Then we define the rescaled mapping f˜n : Sn→M byf˜n(y) = fn(expx(cn+λnσ(y)), (3.22)where σ : S2 \{p+}→ TxM is the stereographic projection from the north pole p+and Sn = {y ∈ S2\{p+} : cn +λnσ(y) ∈ D(0,2εn)}. Note that the disk D(cn,λn)corresponds to the northern hemisphere. The parameters cn,λn are chosen in such29a way that Sn→ S2 \{p−} as n→ ∞, where p− is the south pole.For the metrics on Sn pulled back from hn, { f˜n} is a sequence of harmonic mapsdefined on Sn with uniformly bounded energies. Let S˜ = S{ f˜n} be the bubbling setfor the sequence { f˜n} (defined as in (3.21)). Then, as in the case for { fn}, Theorem2.3 in [45] and the removable singularity theorem imply that { f˜n}, by passing to asubsequence if necessary, converges locally in C1(S2 \ S˜∪{p−},N) to a harmonicmap sx,1 : S2→ N. The harmonic map sx,1 is called a bubble at x.The above procedure is then performed at each y∈ S˜, and this produces bubbleson the bubble sx,1. Iterate this procedure. Note that the process would terminate ata finite number of steps since the energy of each nontrivial bubble is at least ε0.So far, for each x ∈ S, we have associated to it finitely many harmonic mapssx,i : S2x,i→ (N,g), where i= 1,2, · · · ix. Now we describe how the original sequence{ fn} converges to f∞ and the sx,i’s. Due to formation of bubbles, we renormalizethe mapping { fn} in order to formulate C0 convergence. First we restrict each fntoΣ\⋃x∈Sexpx D(cn,εn). (3.23)It is shown (Lemma 1.3 in [42]) that for each x ∈ S, the image of ∂D(cn,εn) underfn always lies in the ball B( f∞(x),C/n) for some positive constant C. Thus we canredefine fn on the disk expx D(cn,εn) by coning off the image at each x ∈ S (see(1.12) in [42]). The resulting map is denoted f¯n ∈C0(Σ,N). Note that f¯n convergesin C0 to f∞.For the first layer of bubbles, we restrict the definition of the rescaled map-ping f˜n (defined in (3.22)) to σ−1(D(cn,nλn)). Again f˜n maps ∂D(cn,nλn) toB( f˜ (p−),C/n) for some C. Hence, as for f¯n, we can similarly cone off the imageof f˜n. The resulting continuous mapping is denoted byR f n,x,1 : S2x,1→ N.Then similarly define R f n,x,i : S2x,i→ N, where i = 2, · · · , ix, for each level of bub-bling.The restriction of fn to the annular region An = D(cn,εn)\D(cn,nλn) is calledthe neck map. The limit of the images of fn|An will connect the base map f¯ and30the bubbling maps sx,1 by joining f∞(x) and sx,1(p−). It is shown in Lemma 2.1 of[42] that the neck maps fn|An converge to points, and there is no energy lost in thislimiting process. The same argument applies to bubbles at all levels of the bubbletree, hence the bubbles are connected to the bubbles at the previous level.Now we define the bubble tower T for the domain of the renormalized map.First of all, for each x ∈ S, we attach a 2-sphere S2, and on this S2 we attach a2-sphere S2 at each y ∈ S˜. Then we repeat this construction on the third layer andso on. Hence we obtain a bubble tower T . Let I be the finite set that indexes allbubbles and 0 ∈ I for Σ. Then the family of maps{ f¯n,R f n,x,i : x ∈ S, i = 1,2, · · · , ix}can be described in a simple notation fn,I : T → N. Let fI : T → N be the family ofmaps given by { f∞,sx,i : x ∈ S, i = 1,2, · · · , ix}.Now we can state the following bubbling convergence theorem:Theorem 3.2.1. (Theorem 2.2 in [42]) Let fn : (Σ,hn)→ (N, g¯) be a sequence ofharmonic maps from a Riemannian surface (Σ,hn) to a compact Riemannian man-ifold (N, g¯) with Ehn,g¯( fn) ≤ E0. Assume in addition that the metrics hn convergeto h, then there is a subsequence (still use the same notation) of { fn} and a bubbletower domain T so that the sequence of renormalized maps{ fn,I : T → N} (3.24)converges in W 1,2∩C0 to a smooth harmonic bubble tree map fI : T → N. More-over,1. (No energy loss) Ehn,g¯( fn) converges to ∑ j∈I E( f j), and2. (Zero distance bubbling) At each bubble point y (at any level of the bubbletree), the image of the respective base map f j and the bubble map fk, wherej,k ∈ I, meet at f j(y) = fk(p−).Consequently, the image of the limit fI : T → N is connected, and the images of theoriginal maps fn : Σ→ N converge pointwisely to the image of fI .31Note that the above theorem is stated only for hn = h for all n in [42], as re-marked in Section 5 in [42], it still holds if the conformal structures represented bythe metrics hn stay in a bounded domain of the moduli space.We are now ready to prove Theorem 1.1.2.Proof of Theorem 1.1.2. Given a sequence of conformally immersed compact La-grangian self-shrinkers Fn : (Σ,hn) → R4 with a uniform area upper bound, byLemma 3.2.1, the images Fn(Σ) lie in a fixed U for all n. By composing with theisometric embedding (U,G)→ (N, g¯) in Lemma 3.2.2, we regard each Fn as a mapwith image in N. As each Fn is a conformal minimal immersion with respect to hnon Σ and G = g¯ on U , by Proposition 3.2.1, {Fn : (Σ,hn)→ (N, g¯)} is a sequenceof harmonic maps. The area µ˜(Fn) of Fn(Σ) in (N, g¯) is given byµ˜(Fn) =∫Σe−|Fn |24 dµF∗n 〈·,·〉.Therefore, as e−|Fn |24 ≤ 1 we haveµ˜(Fn)≤ µ(Fn)< Λ.As Fn : (Σ,hn)→ (N, g¯) is conformal, the area µ˜(Fn) in N is the same as the energy(Proposition 3.2.1): Ehn,g¯(Fn) = µ˜(Fn). Therefore, the energies of the harmonicmappings Fn : (Σ,hn)→ (N, g¯) are also uniformly bounded by Λ. By assumption,the sequence of metrics hn converges to a Riemannian metric h. Hence we canapply the theory of bubble tree convergence of harmonic maps discussed above. Inparticular, by Theorem 3.2.1, the sequence {Fn} converges in the sense of bubbletree to a harmonic mapping F∞ : (Σ,h)→ (N, g¯) and finitely many harmonic map-pings sx,i : S2→ (N, g¯). Since each Fn has image in U , by the C0 convergence ofthe renormalized map in Theorem 3.2.1, both F∞ and sx,i have image in U . In lightof Lemma 3.2.2, F∞ and sx,i are harmonic mappings into (R4,G).Note that some of these mappings, including F∞, might be trivial. Let sx,i :S2→ R4 be a nontrivial bubble. Then we claim that sx,i is a conformally branchedLagrangian self-shrinker. First of all, as sx,i is nontrivial, by Proposition 3.2.1, sx,iis a branched conformal minimal immersion into (R4,G). By Lemma 2.1.1, sx,i is32a conformally branched self-shrinker.It remains to show that sx,i is Lagrangian. When i = 1, sx,1 is the limit of therescaling map f˜n (see (3.22)). Since the rescalings are performed on the domains,each f˜n is Lagrangian since it has the same image in R4 as the Lagrangian immer-sion Fn at the corresponding points. As { f˜n} converges locally uniformly in C1to sx,1 on S2 \ S˜∪{p−}, sx,1 is Lagrangian when restricted to S2 \ S˜∪{p−}. Thesmoothness of sx,1 then implies that sx,1 is indeed Lagrangian on S2. The sameargument applies to bubbles at any level, and so all nontrivial sx,i are Lagrangian inR4.However, according to Theorem 1.1.1, there does not exist any nontrivial con-formally branched Lagrangian self-shrinking immersion of S2 in R4. Therefore,there does not exist nontrivial sx,i. From the construction of the bubbling con-vergence, we conclude that the set S of the bubbling points is empty. Thus theconvergence Fn→ F∞ is in C1(Σ).By Theorem 3.3 in [44], since each Fn : Σ→ N is nontrivial by definition, thereis an ε > 0 such that Ehn,g¯(Fn)≥ ε for all n ∈N. As the convergence Fn→ F∞ is inC1(Σ) (or using (2) in Theorem 3.2.1) and hn→ h,Ehn,g¯(Fn)→ Eh,g¯(F∞) as n→ ∞. (3.25)Hence Eh,g¯(F∞)≥ ε and F∞ is nontrivial. (Alternatively, one can also use the esti-mate of the diameter of each self-shrinker and the Hausdorff convergence as in theproof of Theorem 1.1.3 in the next section to show that F∞ is nontrivial).Again, as the convergence Fn→ F∞ is in C1(Σ) and the metrics hn converge toh smoothly, the harmonic map F∞ : (Σ,h)→ (N, g¯) is also conformal as each Fn is.Thus as a mapping into R4, F∞ is a conformally branched minimal immersion withrespect to the metric G defined by (3.19). The C1 convergence also implies that F∞is Lagrangian in R4. By Lemma 2.1.1, F∞ is a conformally branched Lagrangianself-shrinker in R4.Lastly, by picking a subsequence if necessary, we have the C∞ convergenceof {Fn} as follows. Since Fn : (Σ,hn)→ (N, g¯) is harmonic and {hn} convergessmoothly to h, using the standard elliptic estimates (Chapter 6 of [26]) and a boot-33strapping argument, there are constants C(m) such that||Fn||Cm ≤C(m)for all n ∈ N . Using the Arzela`-Ascoli theorem and picking a diagonal subse-quence, one shows that a subsequence of {Fn} converges smoothly to a smoothmapping Σ→ R4, which has to be F∞. Thus the theorem is proved.3.3 Proof of Theorem 1.1.3Proof. Let {Fn :T→R4} be a sequence of conformally branched Lagrangian self-shrinking tori in R4 with a uniform area upper bound. Let hn be the metric on thetorus T which is conformal to F∗n 〈·, ·〉 and with zero Gauss curvature as discussedin section 1.1. If we can show that hn stays in a bounded domain of the modulispace, then Theorem 1.1.3 follows from Theorem 1.1.2.Using Lemma 3.2.1, Lemma 3.2.2 and Lemma 2.1.1, we regard each Fn as aminimal immersion in (N, g¯), this means that Fn : (T,hn)→ (N, g¯) is conformaland harmonic. Assume the contrary that the conformal structures degenerate. Inthis case, there is a mapping Fˆ∞ from Σ∞ to N and the image Fn(T) converges in theHausdorff distance to Fˆ∞(Σ∞) in N. Here Σ∞ is a stratified surfaces Σ∞ = Σ0∪Σbformed by the principal component Σ0 and bubble component Σb. The princi-pal component Σ0 is formed by pinching several closed, homotopically nontrivialcurves in T and the bubbling component is a union of spheres. There are no necksbetween the components since Fn is conformal. The map Fˆ∞ is continuous on Σ∞and harmonic when restricted to each component of Σ∞. Since all the componentsintersect each others possibly at finitely many points, Fˆ∞ is harmonic except at afinite set Sˆ.Since the conformal structures determined by the metrics hn degenerate, at leastone homotopically nontrivial closed curve must be pinched to a point as n→ ∞.It follows that Σ0 is a finite union of S2’s. Each of these 2-spheres is obtained byadding finitely many points to the cylinder S1×R that comes from pinching one ortwo homotopically nontrivial loops: two at the infinity and at most finitely many atthe blowup points of the sequence Fn, again by (4) in Proposition 3.2.2. Therefore,Fˆ∞ is a finite union of harmonic mappings Fˆ i∞ from the sphere to N.34Since all Fn are conformal, there are no necks between the components. Thebubble tree convergence described above are given by the results in [13] or [10]. In[13], the limiting surface is a stratified surface with geodesics connecting the twodimensional components. But together with conformality of each Fn and Propo-sition 2.6 in [13], one sees that all the geodesics involved have zero length. Al-ternatively, we can use the compactness theorem in [10] for Σ = T: Suppose that{ fk} is a sequence of W 2,2 branched conformal immersions of (T,hk) in a compactRiemannian manifold M. Ifsupk{µ( fk)+W ( fk)}<+∞where W ( fk) = 14∫ |Hk|2, then either { fk} converges to a point, or there is a strati-fied sphere Σ∞ and a W 2,2 branched conformal immersion f∞ : Σ∞→M, such thata subsequence of { fk(Σ)} converges to f∞(Σ∞) in the Hausdorff topology, and thearea and the Willmore type energy satisfyµ( f∞) = limk→+∞µ( fk) and W ( f∞)≤ limk→+∞W ( fk).The conditions of the theorem are satisfied by the sequence {Fn} as the area µ˜(Fn)and the Willmore energy W (Fn) in N (which is zero as each Fn is minimal immer-sion in N) are uniformly bounded. In our situation, Fk will not converges to a pointsince diam(Fn(T )) ≥ 2 by equations (3.18) and (3.17). Thus the inequality on theWillmore type energies in NW (Fˆ∞)≤ limn→∞W (Fn) = 0implies that the limiting S2’s are all branched minimal surfaces in N.Consequently, the images Fn(T) converge in the Hausdorff distance to the im-age of finitely many harmonic maps S2→ N. These harmonic maps are branchedconformal immersions, which are also Lagrangian by similar reasons as in theproof of Theorem 1.1.2. By Theorem 1.1.1, all these harmonic maps are trivial.Hence, the images Fn(T) converge in the Hausdorff distance to a point in R4. Thisis impossible by the diameter estimate diam(Fn(T))≥ 2. This contradiction shows35that the conformal structures cannot degenerate and that finishes the proof of The-orem 1.1.3.3.4 Lagrangian self-shrinking tori with small areaIn this section, we restrict to Lagrangian tori and prove Theorem 1.1.4. We will usea contradiction argument. By doing so we need the compactness theorem 1.1.3 anda factorization result of Lamm and Scha¨tzle [32] concerning branched conformalimmersions of tori into R4 with Willmore energy 8pi .The proof of Theorem 1.1.4 will be divided into the following results. Werecall that XΛ stands for the space of branched conformally immersed Lagrangianself-shrinking tori of area no larger than Λ.Proposition 3.4.1. There is a positive number ε0 so that if F ∈ X32pi+ε0 , then F isimmersed.Proof. Arguing by contradiction, we assume that there is a sequence Fn : T→ R4of branched conformal Lagrangian self-shrinking tori so thatliminfn→∞ µ(Fn)≤ 32pi (3.26)and each Fn has a nonempty set of branch points. Using Theorem 1.1.3, by passingto a subsequence if necessary, the sequence {Fn} converges smoothly to a branchedconformal Lagrangian self-shrinking torus F∞ :T→R4. Let Bn be the set of branchpoints of Fn. Since T is compact, again by passing to a subsequence if necessary,there is a sequence {pn}, where pn ∈ Bn for each n ∈ N, so that pn→ p ∈ T. AsDFn(pn) = 0 for all n ∈N and the convergence Fn→ F∞ is smooth, DF(p) = 0 andso p is a branch point of F∞, where DF,DFn are the differentials of F,Fn, respec-tively. By the theorem of Li and Yau (Theorem 6 in [36], see also the appendix in[31] for the generalization to branched immersions), since F∞ is not embedded,W (F∞)≥ 8pi. (3.27)36On the other hand, from (3.26) and Theorem 1 in [10],µ(F∞)≤ liminfµ(Fn)≤ 32pi.Together with (3.27) and (3.17) we have W (F∞) = 8pi . Since F∞ has a branchpoint, Proposition 2.3 in [32] implies that F∞ factors through a branched conformalimmersion g : T→ S2. It follows that there is a branched conformal Lagrangianself-shrinking sphere h : S2→ R4 so that F∞ = h◦g. However, by Theorem 1.1.1,such a mapping h does not exist. This contradicts the existence of the sequence{Fn}. The proposition is now proved.Proposition 3.4.1 and Theorem 1.1.3 lead toTheorem 3.4.1. Let ε0 be as in Proposition 3.4.1. Then the space of all Lagrangianimmersed self-shrinking tori with area less than or equal to 32pi+ ε0 is compact.Next we prove part (2) in Theorem 1.1.4.Corollary 3.4.1. (Curvature Estimates) There is C0 > 0 so that if F : T→ R4is a Lagrangian immersed self-shrinking torus with area less than or equals to32pi+ ε0, then the second fundamental form of F is bounded by C0.Proof. Assume this were not true. Then there is a sequence Fn : T→ R4 of La-grangian immersed self-shrinking tori with area less than 32pi+ ε0 so thatmaxFn(T)|An| → ∞, (3.28)where An is the second fundamental form of the immersion Fn. Using Theorem3.4.1, a subsequence of {Fn} converges smoothly to an immersed self-shrinker F∞.In particular, we have(gn)i j =∂Fn∂xi· ∂Fn∂x j−→ ∂F∞∂xi· ∂F∞∂x j= (g∞)i j, as n→ ∞.Since g∞ is positive definite as F∞ is immersed, there is a positive number C so thatgn ≥Cδi j for all n. So g−1n are uniformly bounded. HencemaxFn(T)|An|2 = maxFn(T)gi jn gkln 〈(An)ik,(An) jl〉37are uniformly bounded and (3.28) is impossible.To finish the proof of Theorem 1.1.4, it remains to prove the first part in (1).Proposition 3.4.2. There is a positive constant ε1 ≤ ε0 so that if F ∈X32pi+ε1 , thenF is embedded.Proof. As in the proof of Corollary 3.4.1, assume the contrary that there is asequence {Fn} of immersed, non-embedded Lagrangian self-shrinking tori withµ(Fn) ≤ 32pi + ε0 and µ(Fn)→ 32pi . By Theorem 3.4.1, after passing to a sub-sequence if necessary, {Fn} converges smoothly to an immersed Lagrangian self-shrinking torus F∞ : T→ R4 with area µ(F∞) = 32pi . By (3.17), the Willmoreenergy of F∞ is 8pi . Since each Fn is non-embedded, there are distinct pointspn,qn ∈ T so thatFn(pn) = Fn(qn). (3.29)As T is compact, we may assume pn → p and qn → q. Taking n→ ∞ in (3.29),we have F∞(p) = F∞(q). First of all, we must have p = q: Indeed, if p 6= q, thenF∞ is not embedded and that contradicts Theorem 2.2 in [32], which states that anyimmersion F : T→ R4 with W (F) = 8pi has to be embedded.Let dn be the distance function on T induced by the pullback metric F∗n 〈·, ·〉. Asp = q and {Fn} converges smoothly to F , we have `n := dn(pn,qn)→ 0 as n→ ∞.Let ηn : [0, `n]→ T2 be a shortest geodesics in (T,F∗n 〈·, ·〉) joining pn to qn. SinceFn(ηn(0)) = Fn(ηn(`n)), Fn ◦ηn : [0, `n]→ R4 is a closed curve in R4 with length`n. Let γn : [0, `n]→R4 be the translation γn(t) = Fn ◦ηn(t)−Fn(pn). Then each γnis parameterized by arc length and γn(0) = γn(`n) = 0 ∈ R4. Using the following38simple estimates`n =∫ `n0〈γ ′n(t),γ ′n(t)〉dt=−∫ `n0〈γn(t),γ ′′n (t)〉dt+ 〈γn(`n),γ ′n(dn)〉−〈γn(0),γ ′n(0)〉=−∫ `n0〈γn(t),γ ′′n (t)〉dt≤∫ `n0|γn(t)| · |γ ′′n (t)|dt≤ `n∫ `n0|γ ′′n (t)|dt,we obtain ∫ `n0|γ ′′n (t)|dt ≥ 1.Since `n→ 0, the above inequality implies that there is sn ∈ [0, `n] so that |γ ′′n (sn)|→∞ as n→ ∞. Since ηn is a geodesic on (T,F∗n 〈·, ·〉),γ ′′n = (Fn ◦ηn)′′ = ∇nη ′nη ′n+An(η ′n,η ′n) = An(η ′n,η ′n),where ∇n is the Levi-Civita connection on (T,F∗n 〈·, ·〉) and An is the second funda-mental form of Fn(T) in R4. Thus|γ ′′n (t)| ≤ |An(ηn(t))|and this impliesmaxFn(T)|An|2→ ∞as n→ ∞. However, this is impossible by Corollary 3.4.1.39Chapter 4Finiteness of entropy andpiecewise Lagrangian meancurvature flow4.1 A Łojasiewicz-Simon type gradient inequality forbranched self-shrinking toriIn the last chapter we showed that with a small area bound, all Lagrangian self-shrinking tori are embedded. This makes it much easier to study the space X32pi+ε0 ,as all nearby Lagrangian self-shrinking tori can be deformed to each other by usingthe normal vectors fields. However, it is difficult in general to relate two nearbybranched conformal immersions, even if they are Ck-close when treated as map-pings to the Euclidean space. In particular, it seems difficult to extend the pertur-bation procedure as in [17, 34], where the stability condition is described by usingthe normal vector fields, to branched conformal self-shrinkers.In this section, we show that the entropy λ is locally a constant function in thespace of branched conformal compact self-shrinking tori F : T→ R4. To do this,we derive a Łojasiewicz-Simon type gradient inequality for branched conformalself-shrinking tori F : T→ R4. In the genus one case, the explicit expression ofthe conformal structures in the Teichmu¨ller space makes the computation and the40real analyticity of the functional E transparent. Once this is done, together withthe compactness of XΛ, we conclude the proof of Theorem 1.2.1.4.1.1 A Fredholm operator of index zeroLet (Σ,g) be a compact Riemannian surface and (M,h) a Riemannian manifold.Given a C1 mapping F : Σ→M, the energy of F is given byEg,h(F) =12∫Σeg,h(F)dµg,where eg,h(F) is the norm of the differential DFx : TxΣ→ TF(x)M. Locally it isgiven byeg,h(F) = gi jhαβ∂Fα∂xi∂Fβ∂x j.For a fixed h, define E : C1(Σ,M)×{g : g is a Riemannian metric on Σ} → RbyE (F,g) = Eg,h(F).Lemma 4.1.1. If F : (Σ,g)→ (M,h) is conformal, then g is a critical point of Ewith respect to all its smooth variations gs, where g0 = g. That is,ddsEgs,h(F)∣∣∣∣s=0= 0.Proof. Let gs be a family of smooth metrics on Σ so that g0 = g and g˙ = dds gs∣∣s=0.Thendds(gi j√detg)∣∣∣∣s=0=−gikg jl g˙kl√detg+12gi j√detggkl g˙kl=(12gklgi j−gikg jl)g˙kl√detgThusddsegs,h(F)dµgs∣∣∣∣s=0=(12gklgi j−gikg jl)g˙klhαβ∂Fα∂xi∂Fβ∂x jdµg (4.1)41Since F is conformal,hαβ∂Fα∂xi∂Fβ∂x j= ϕgi j (4.2)for some function ϕ on Σ. Put (4.2) into (4.1) and use gi jgi j = 2 since Σ is twodimensional, we see that ddsEgs,h(F)∣∣s=0 = 0, as claimed.On the other hand, recall that a branched minimal immersion is (weakly) con-formal and harmonic, and we have the following ([45], Theorem 1.8)Proposition 4.1.1. If u is critical map of E with respect to the variations of u andthe conformal structures on Σ, then u is a branched minimal immersion.Let U be an open subset in the upper half space H = {τ ∈ C | Imτ > 0}. Itis well-known that the upper half space represents the Teichmu¨ller space of thestandard torus T = R2/{1, i} and we treat U as a local parameterization of theconformal structures on T near a given one.Let 0 < α < 1 be fixed. DefineU =C2,α(T,R4)×U,C k,α =Ck,α(T,R4)⊕R2W k,p =W k,p(T,R4)⊕R2L 2 =W 0,2.Note that C k,α ,W k,α are Banach spaces 1 with the norms‖(φ ,ν)‖k,α = ‖φ‖Ck,α + |v|,‖(φ ,ν)‖W k,p = ‖φ‖W k,p + |v|respectively. When (M,h) = (R4,G), where G is as in (2.3), the functional E :U → R takes the formE (u,τ) =12∫Te−|u|24 |Du|2τdµτ , (u,τ) ∈U . (4.3)1All Banach spaces considered in this thesis are real Banach spaces.42Here gτ is the metric on T given bygτ =(1 τ10 τ2)T (1 τ10 τ2)(4.4)anddµτ = dµgτ =√detgτ dxdy, |Du|2τ = gi jτ Diu ·D ju.The metric gτ is in the conformal class represented by τ , as it can be seen easily thatgτ is the pullback metric via the linear mapping from T= R2/{1, i} to R2/{1,τ}.Note that for each fixed τ , E (·,τ) is the Dirichlet energy functional of the mappingsu : (T,gτ)→ (R4,G).By Lemma 2.1.1, minimal surfaces in (R4,G) corresponds to self-shrinkingsurfaces in R4. Thus Lemma 4.1.1 and Proposition 4.1.1 imply the followingProposition 4.1.2. (u,τ) is a critical point of E if and only if u : (T,gτ)→R4 is abranched conformal self-shrinking torus.Next we consider the L2-gradient M : U → C 0,α of E . That is, we find foreach (u,τ) ∈U an elementM (u,τ) ∈ C 0,α so that for all (φ ,ν) ∈ C 2,α ,dds∣∣∣∣s=0E (u+ sφ ,τ+ sν) = 〈M (u,τ),(φ ,ν)〉u,τ . (4.5)Here we define〈φ1,φ2〉u,τ =∫Tφ1 ·φ2 e−|u|24 dµτand〈(φ1,ν1),(φ2,ν2)〉u,τ = 〈φ1,φ2〉u,τ +ν1 ·ν2. (4.6)Integrating by parts, we see thatM (u,τ) =(−gi jτ e|u|24 D j(e−|u|24 Diu)− 14 |Du|2τu,∇Euτ)(4.7)where E u : U →R is given by E u(τ) = E (u,τ) and ∇E uτ is the gradient of E u at τ .Let (u,τ) be a critical point of E , that is,M (u,τ) = 0. LetL =L(u,τ) : C2,α → C 0,α43be the Fre´chet derivative ofM at (u,τ). We will show thatL (φ ,ν) =(Lφ +∇νB,(∇2E uτ )ν+ 〈∇Bτ ,φ〉u,τ)(4.8)whereLφ =−gi jτ e|u|24 D j(e−|u|24 Diφ)− 14 |Du|2τφ +12gi jτ D j(u ·φ)Diu− 12gi jτ (D ju ·Diφ)u(4.9)and ∇2E uτ is the Hessian of E u at τ; furthermore, B : U →C0,α(T,R4) is given byB(σ) =−gi jσ(e|u|24 D j(e−|u|24 Diu)+14(Diu ·D ju)u)(4.10)and ∇Bτ denotes the Fre´chet derivative of B at τ and ∇νBτ stands for the Fre´chetderivative of B at τ in the direction ν :∇νBτ =dds∣∣∣∣s=0B(τ+ sν).To derive (4.8), note that the two terms in the first component of (4.8) arise fromdirect differentiation of the first component of (4.7) with respect to φ and ν . Toderive the second component, note that (∇2E uτ )ν is just the directional derivativeof ∇E uτ with respect to ν . Thus we need to show that ∇φ∇E uτ = 〈∇Bτ ,φ〉u,τ , where∇φ∇E uτ =dds∣∣∣∣s=0∇E u+sφτ .Note∇E uτ =12∫T(∇gi jτ )e−|u|24 (Diu ·D ju)dµτ + 14tr(g−1τ ∇gτ)E (u,τ),where the second term on the right comes from differentiating the volume formdµτ . Since (u,τ) is a critical point of E , this term vanishes when we differentiate44with respect to φ . Using this observation and integration by parts,∇φ∇E uτ =12∫T(∇gi jτ )∇φ(e−|u|24 (Diu ·D ju))dµτ=12∫T(∇gi jτ )(−12(u ·φ)e− |u|24 (Diu ·D ju)+2e−|u|24 (Diu ·D jφ))dµτ=−∫T(∇gi jτ )(14(Diu ·D ju)u+ e|u|24 D j(e−|u|24 Di) ·φ)e−|u|24 dµτ=∫T∇Bτ ·φ e−|u|24 dµτ= 〈∇Bτ ,φ〉u,τ .Thus (4.8) is shown.Lemma 4.1.2. Let (u,τ) be a critical point of E . For all (φ ,ν),(ψ,η) ∈ C 2,α , wehave〈L (φ ,ν),(ψ,η)〉u,τ = 〈(φ ,ν),L (ψ,η)〉u,τ . (4.11)Proof. Let φ ,ψ ∈C2,α(T,R4), then from (4.8) and (4.10),〈Lφ ,ψ〉u,τ = 〈gi jτ Diφ ,D jψ〉u,τ − 14〈|Du|2τφ ,ψ〉u,τ+12∫gi jτ D j(u ·φ)Diu ·ψ e−|u|24 dµτ − 12∫gi jτ (D ju ·Diφ)(u ·ψ)e−|u|24 dµτ .(4.12)Integrating by parts for the third term on the right hand side in (4.12) gives12∫gi jτ D j(u ·φ)Diu ·ψ e−|u|24 dµτ=−12∫gi jτ (u ·φ)(Diu ·D jψ)e−|u|24 dµτ − 12∫(u ·φ)gi jτ D j(e−|u|24 Diu) ·ψ dµτ .(4.13)SinceM (u,τ) = 0, we have by (4.7)gi jτ D j(e−|u|24 Diu) =−14e− |u|24 |Du|2τu.45Putting this into (4.13), we have〈Lφ ,ψ〉u,τ = 〈gi jτ Diφ ,D jψ〉u,τ − 14〈|Du|2τφ ,ψ〉u,τ− 12∫gi jτ (u ·φ)(Diu ·D jψ)e−|u|24 dµgτ −12∫gi jτ (D ju ·Diφ)(u ·ψ)e−|u|24 dµτ+18∫(u ·φ)(u ·ψ)|Du|2τe−|u|24 dµτ .Note that the right hand side is symmetric in φ and ψ . Thus〈Lφ ,ψ〉u,τ = 〈φ ,Lψ〉u,τ , ∀φ ,ψ ∈C2,α(T,R4). (4.14)Using this, we have〈L (φ ,ν),(ψ,η)〉u,τ = 〈Lφ +∇νBτ ,ψ〉u,τ +(∇2E uτ ν+ 〈∇Bτ ,ψ〉u,τ) ·η= 〈Lφ ,ψ〉u,τ + 〈∇νBτ ,ψ〉u,τ + 〈∇ηBτ ,φ〉u,τ +(∇2E uτ ν) ·ηAgain, the right hand side is symmetric in (φ ,ν) and (ψ,η). We can now concludethe proof of the lemma.Remark 1. Note that the apparent self-adjointness expression forL in (4.11) onlyholds in C 2,α , and L is an operator from C 2,α to C 0,α . Nevertheless, (4.11) isuseful in proving the following theorem.Theorem 4.1.1. L is a Fredholm operator of index zero at a critical point (u,τ)of E .Proof. The proof will be divided into several steps.Step 1. We show that dimkerL is finite.Consider the first component ofL ,Lφ +∇νBτ = 0. (4.15)This equation is bilinear in φ ,ν . Let S be the subspace of R2 so that ν ∈ S if andonly if (4.15) has a solution. If S = {(0,0)}, then dimkerL = dimkerL <∞ sinceL is elliptic. If not, let {νi} be a basis of S. Pick φi ∈C2,α(T,R4) so that φi satisfies46(4.15) with ν = νi. Let (φ ,ν) ∈ kerL . Then ν ∈ S. Write ν = ∑i siνi for somesi ∈ R. Then φ − siφi ∈ kerL and thus(φ ,ν) = (φ0,0)+∑isi(φi,νi)for some φ0 ∈ kerL. Again, due to the ellipticity of L, dimkerL is finite, hencekerL is finite dimensional.Step 2. L has finite dimensional cokernel. Moreover, dimkerL = dimcokerL .We will show that the mappingkerL ↪→ C 2,α ↪→ C 0,α pi→ cokerL (4.16)is bijective, where pi is the projection to the quotient cokerL = C 0,α/ImL .Firstly, if (ψ1,η1),(ψ2,η2)∈ kerL represent the same element in cokerL, thenthere is (φ ,ν) ∈ C 2,α so that(ψ,η) := (ψ1−ψ2,η1−η2) =L (φ ,ν).Using (4.11),〈(ψ,η),(ψ,η)〉u,τ = 〈L (φ ,ν),(ψ,η)〉u,τ = 〈(φ ,ν),L (ψ,η)〉u,τ = 0.Thus (ψ,η) = 0 and so the mapping kerL → cokerL defined in (4.16) is injec-tive.Secondly, we show that the mapping kerL → cokerL is surjective. Let ImLbe the L2 closure of the image ofL inL 2 with respect to the inner product definedin (4.6). Let (ψ,η) ∈ C 0,α represents an element in cokerL . We decompose(ψ,η) into the component in ImL and ImL ⊥. That is,(ψ,η) = (ψ>,η>)+(ψ⊥,η⊥) (4.17)for some ψ>,ψ⊥ ∈ L2(T,R4). Note that〈(ψ⊥,η⊥),L (φ ,ν)〉u,τ = 047for all (φ ,ν) ∈ C 2,α . Letting ν = 0 and using (4.8), we have〈Lφ ,ψ⊥〉u,τ + 〈∇η⊥Bτ ,φ〉u,τ = 0, ∀φ ∈C2,α(T,R4).Note that the above equation is of the form∫(−gi jτ Di jφ +A iDiφ +Bφ) ·ψ⊥dxdy =∫F ·φ dxdy,where A i = (A iβγ) and B = (Bβγ) are (4× 4)-matrix-valued smooth functionsand F = (Fβ ) is a R4-valued smooth function. If we choose φ = (ρ,0,0,0),where ρ ∈C∞(T,R), we have∫(−gi jτ Di jρ+A i11Diρ+B11ρ)ψ⊥1 dxdy = D(ρ), (4.18)whereD(ρ) =−∫∑k 6=1A ik1ψ⊥k Diρ dxdy−∫∑k 6=1Bk1ψ⊥k ρ dxdy+∫F1ρ dxdy. (4.19)Since ψ⊥k are in L2 (noting that the L2 spaces with respect the area elementse−|u|24 dµτ and dxdy coincide over T), as a distribution, D is in H−1loc . Thus theElliptic Regularity Theorem (Theorem 6.33 in [24]) asserts ψ⊥1 ∈ H1loc. Similarly,we have ψ⊥k ∈ H1loc for k = 2,3,4. Putting this information into (4.19), we see thatD ∈ H0loc, and in turn, this implies φ⊥1 ∈ H2loc by the Elliptic Regularity Theoremagain. By a standard bootstrapping argument and the Sobolev embedding theo-rem, we see that ψ⊥ ∈ C2,α (in fact, smooth). Using (4.11) and the definition of(ψ⊥,η⊥), we have(L (ψ⊥,η⊥),(φ ,ν))u,τ = 0for all (φ ,ν) ∈ C 2,α , thusL (ψ⊥,η⊥) = 0.The smoothness of (ψ⊥,η⊥) asserts (ψ>,η>) ∈ C 0,α . If we can show that(ψ>,η>) ∈ ImL , (4.20)48then pi(ψ,η) = pi(ψ⊥,η⊥) by (4.17) and it follows that the mapping kerL →cokerL defined in (4.16) is surjective and we are done. To show (4.20), recall that(ψ>,η>) ∈ ImL . Thus there is a sequence (φn,νn) ∈ C 2,α so that L (φn,νn)→(ψ>,η>) inL 2. Using theL 2 inner product, we decompose (φn,νn) into(φn,νn) = (φKn ,νKn )+(φPn ,νPn ), (4.21)where (φKn ,νKn ) ∈ kerL and (φPn ,νPn ) ∈ kerL ⊥. Then by setting(ψn,ηn) =L (φPn ,νPn ) (4.22)and usingL (φKn ,νKn ) = 0, we have(ψn,ηn) =L (φPn ,νPn )=L (φn,νn)L 2→ (ψ>,η>).The convergence above in particular implies that ‖ψn‖L2 ≤C for some constant C.From the first component of (4.8), which isLφPn = ψn−∇νPn Bτ ,the standard elliptic estimates (Theorem 9.11 in [26]) implies that there are con-stants C′,C′′,C′′′ > 0 so that‖φPn ‖W 2,2 ≤C′(‖φPn ‖L2 +‖ψn−∇νPn Bτ‖L2)≤C′ (‖φPn ‖L2 +‖ψn‖L2 +C′′|νPn |)≤C′′′ (‖(φPn ,νPn )‖L 2 +1) .(4.23)Next, we show that the sequence {‖(φPn ,νPn )‖L 2} is bounded. Assume not,then by taking a subsequence if necessary, we have ‖(φPn ,νPn )‖L 2 → ∞. Let(φ˜n, ν˜n) =(φPn ,νPn )‖(φPn ,νPn )‖L 2. (4.24)49Then, as (ψn,ηn) converges to (ψ>,η>) inL 2,L (φ˜n, ν˜n) =(ψn,ηn)‖(φPn ,νPn )‖L 2L 2−→ 0. (4.25)Since ‖(φ˜n, ν˜n)‖L 2 = 1, we may assume ν˜n→ ν˜ for some ν˜ ∈R2. From (4.23) and(4.24), the sequence {‖φ˜n‖W 2,2} is bounded. Hence, again by taking subsequenceif necessary, there is φ˜ ∈W 2,2(T,R4) so that φ˜n→ φ˜ in W 1,2(T,R4). Using (4.25),we have Lφ˜ +∇ν˜Bτ = 0 weakly in W 1,2(T,R4),∇2E uτ ν˜+ 〈∇Bτ , φ˜〉u,τ = 0Since φ˜ ∈W 2,2(T,R4), the first equation is actually satisfied strongly in W 2,2(T,R4).Since ∇ν˜Bτ is smooth, by the elliptic regularity, φ˜ is smooth. Thus (φ˜ , ν˜) ∈C 2,α and L (φ˜ , ν˜) = 0, in other words, (φ˜ , ν˜) ∈ kerL . On the other hand, since(φ˜n, ν˜n)→ (φ˜ , ν˜) in L 2 and (φ˜n, ν˜n) ∈ kerL ⊥, we also have (φ˜ , ν˜) ∈ kerL ⊥.Thus (φ˜ , ν˜) = (0,0). But this is impossible as ‖(φ˜ , ν˜)‖L 2 = 1 since (φ˜n, ν˜n)→(φ˜ , ν˜) inL 2 and ‖(φ˜n, ν˜n)‖L 2 = 1. The contradiction leads to the conclusion thatthe sequence {‖(φPn ,νPn )‖L 2} is bounded.From (4.23), the sequence {‖(φPn ,νPn )‖W 2,2} is also bounded. By taking a sub-sequence if necessary, there is (φ ,ν) ∈ W 2,2 so that (φPn ,νPn )→ (φ ,ν) in W 1,2andL (φ ,ν) = (ψ>,ν>).The first component of this is given byLφ +∇νBτ = ψ>.Since φ ∈W 2,2(T,R4) and ψ> ∈C0,α(T,R4), the standard elliptic regularity (The-orem 9.19 in [26]) implies that φ ∈C2,α(T,R4). Thus (φ ,ν) ∈ C 2,α . This shows(ψ>,η>) ∈ ImL . Therefore, the mapping kerL → cokerL is surjective.Step 3. From the previous two steps, the bounded operator L has finite di-mensional kernel and cokernel so it is a Fredholm operator ofindexL = dimkerL −dimcokerL = 0.50This completes the proof of the theorem.4.1.2 A Łojasiewicz-Simon type inequalityNext we prove a Łojasiewicz-Simon gradient inequality for compact branched self-shrinkers F : T→ R4. As in [49], we use the Liapunov-Schmidt reduction argu-ment and the classical Łojasiewicz inequality in [37]. See [23] for a Łojasiewicz-Simon inequality in the abstract setting and the related work in the reference therein.LetΠ :L 2→ kerLbe the L2-projection with respect to the L2 inner product:〈(ψ1,ν1),(ψ2,ν2)〉L 2 =∫Tψ1 ·ψ2 dxdy+ν1 ·ν2 (4.26)for all (ψ1,ν1),(ψ2,ν2) ∈L 2. Recall that kerL is a finite dimensional subspaceand kerL ⊂ C ∞. For all k = 0,1,2, · · · , we letΠk : C k,α → C 0,αbe the restriction of Π to C k,α composed with the inclusion kerL ↪→ C 0,α .Lemma 4.1.3. Πk : C k,α → C 0,α is a bounded linear operator for all nonnegativeintegers k. In particular, there is a positive constant Cα so that‖Πk(ψ,ν)‖0,α ≤Cα‖(ψ,ν)‖k,α (4.27)for all (ψ,ν) ∈ C k,α .Proof. Let (χ1,ν1), · · · ,(χn,νn) ∈ kerL be an orthonormal basis of the finite di-mensional space kerL with respect to the inner product in (4.26). Then for any(ψ,ν) ∈L 2, we haveΠ(ψ,ν) =n∑i=1〈(χi,νi),(ψ,ν)〉L 2(χi,νi).51Then we have‖Πk(ψ,ν)‖0,α ≤n∑i=1|〈(χi,νi),(ψ,ν)〉L 2 |‖(χi,νi)‖0,α≤(n∑i=1‖(χi,νi)‖0,α)‖(ψ,ν)‖L 2 .Note that we used the Cauchy-Schwarz inequality and that ‖(χi,νi)‖L 2 = 1. Since∫Tdxdy = 1,we have‖(ψ,ν)‖L 2 ≤maxT2|ψ|+ |ν | ≤ ‖(ψ,ν)‖k,α (4.28)for all nonnegative k. Now (4.27) follows with Cα = ∑ni=1 ‖(χi,νi)‖0,α .To simplify notations, in the sequel we use x,y and a,b to denote elements inC 2,α and C 0,α respectively. Let xc = (u,τ) be a critical point of E as before, thatisM (xc) = 0.Consider the mappingN :U → C 0,α given byN (x) =M (x)+Π2(x− xc). (4.29)Since Π2 is linear, the differential DN at xc is given byDNxc =L +Π2. (4.30)Lemma 4.1.4. DNxc is bijective and its inverse is bounded.Proof. First we show that DNxc is injective. Let DNxc(x) = 0. Then by (4.30) wehaveL (x) =−Π2x.Using (4.11), for all y ∈ kerL we have〈Π2x,y〉u,τ =−〈L x,y〉u,τ =−〈x,L y〉u,τ = 0.52This means that Π2x ∈ kerL is orthogonal to kerL . Therefore, Π2x = 0. ThusL x = 0 and so x ∈ kerL . Hence x =Π2x = 0 and DNxc is injective.By Theorem 4.1.1, L is a Fredholm operator of index zero. Since Π2 isbounded with a finite dimensional range, Π2 is a compact operator and DNxc :C 2,α → C 0,α is Fredholm with index zero (Theorem 5.10 in [46]). Together withthe fact that DNxc is injective, DNxc is also surjective. Finally, the bounded inversetheorem (Theorem 3.8 in [46]) asserts that DNxc has a bounded inverse.By the inverse function theorem for Banach spaces (Theorem 15.2 in [21]),since N is C1 (N is even analytic: see the appendix), there are open neighbour-hoods U1 of xc in U and V1 of 0 in C 0,α so thatN :U1→ V1 is invertible with aC1 inverse Ψ. By shrinking U1,V1 if necessary, we assume that V1 is convex, U1is contained in a convex set U2 ⊂U and (since M and Ψ are C1) there exist twopositive constants M1,M2 so that‖DΨ(a)‖op ≤M1, ∀a ∈ V1,‖DM (x)‖op ≤M2, ∀x ∈U2,(4.31)where ‖ · ‖op denotes the operator norm for the corresponding operator. Using theFundamental Theorem of Calculus, the above imply‖Ψ(a)−Ψ(b)‖2,α ≤M1‖a−b‖0,α (4.32)for all a,b ∈ V1 and‖M (x)−M (y)‖0,α ≤M2‖x− y‖2,α (4.33)for all x,y ∈U1.A main technical result in this section is the following Łojasiewicz-Simon typegradient inequality:Theorem 4.1.2. There is an open neighbourhoodW0⊂U of xc, a positive constantC2 and a constant θ ∈ (0,1/2) depending on E and xc so that|E (x)−E (xc)|1−θ ≤C2‖M (x)‖0,α , ∀x ∈W0. (4.34)53Proof. SinceΠ0 is bounded, there is an open neighbourhood V0 of 0 so that V0,Π0V0⊆V1. For all a ∈ V0, Π0a ∈ V1. Since U2 is convex, the line segment joining Ψ(a)and Ψ(Π0a) is in U2. The Fundamental Theorem of Calculus and (4.5) yieldE (Ψ(a))−E (Ψ(Π0a)) =−∫ 10ddt(E (Ψ(a)+ t(Ψ(Π0a)−Ψ(a)))dt=−∫ 10〈M (Ψ(a)+ t(Ψ(Π0a)−Ψ(a))),Ψ(Π0a)−Ψ(a)〉ut ,τt dt,where we write(ut ,τt) =Ψ(a)+ t(Ψ(Π0a)−Ψ(a)).Using the Cauchy-Schwarz inequality, (4.28), (4.33) and |t| ≤ 1,|E (Ψ(a))−E (Ψ(Π0a))|≤ ‖M (Ψ(a)+ t(Ψ(Π0a)−Ψ(a)))‖L 2‖Ψ(Π0a)−Ψ(a)‖L 2≤ ‖M (Ψ(a)+ t(Ψ(Π0a)−Ψ(a)))‖0,α‖Ψ(Π0a)−Ψ(a)‖2,α≤ (‖M (Ψ(a)‖0,α +M2 t ‖Ψ(Π0a)−Ψ(a)‖2,α)‖Ψ(Π0a)−Ψ(a)‖2,α≤ (‖M (Ψ(a)‖0,α +M2‖Ψ(Π0a)−Ψ(a)‖2,α)‖Ψ(Π0a)−Ψ(a)‖2,α(4.35)On the order hand, since a,Π0a ∈ V1, by (4.32) we have‖Ψ(Π0a)−Ψ(a)‖2,α ≤M1‖Π0a−a‖0,α . (4.36)Using the definition ofN ,Ψ and Π0Π2 =Π2,a =N (Ψ(a)) =M (Ψ(a))+Π2(Ψ(a)− xc) (4.37)Π0a−a =Π0a−M (Ψ(a))−Π2(Ψ(a)− xc)=Π0(a−Π2(Ψ(a)− xc))−M (Ψ(a)). (4.38)Since Π0 is bounded by Lemma 4.1.3,‖Π0(a−Π2(Ψ(a)− xc))‖0,α ≤Cα‖a−Π2(Ψ(a)− xc)‖0,α=Cα‖M (Ψ(a))‖0,α ,54where in the last line we use (4.37) again. Combining this with (4.36) and (4.38),we are led to‖Ψ(Π0a)−Ψ(a)‖2,α ≤C1‖M (Ψ(a))‖0,α (4.39)for all a ∈ V0 with C1 = M1(Cα +1). Putting this into (4.35), we have|E (Ψ(a))−E (Ψ(Π0a))| ≤C3‖M (Ψ(a))‖20,α (4.40)for all a ∈ V0 and for some C3 > 0.Let f : V1∩kerL → R be defined byf (a) = E (Ψ(a)). (4.41)It is easy to show that E , M are analytic (a proof is given in the appendix forcompleteness). Since Π2 is linear,N =M +Π2−Π2(xc)is analytic as well. Hence Ψ is analytic by the analytic version of inverse functiontheorem (Theorem 15.3 in [21]). Consequently, as a composition of analytic func-tions, f is also analytic, and it is defined on an open set in kerL , which is finitedimensional. The classical Łojasiewicz inequality [37] then implies that there is anopen neighbourhood V2 ⊂ V0, constants c > 0 and θ ∈ (0,1/2) so that| f (ξ )− f (0)|1−θ ≤ c| f ′(ξ )|, ∀ξ ∈ V2∩kerL . (4.42)Using (4.41) and (4.5), for all b ∈ V1∩K we havef ′(b)(·) = 〈M (Ψ(b)),DΨb(·)〉u,τ .55Using (4.28), (4.33) and (4.39),| f ′(Π0a)| ≤M1‖M (Ψ(Π0a))‖L 2≤M1‖M (Ψ(Π0a))‖0,α≤M1(‖M (Ψ(Π0a))−M (Ψ(a))‖0,α +‖M (Ψ(a))‖0,α)≤M1(M2‖Ψ(Π0a)−Ψ(a)‖2,α +‖M (Ψ(a))‖0,α)≤C4‖M (Ψ(a))‖0,α(4.43)for some C4 > 0. Now let W0 = Ψ(V2). Thus for every x ∈ W0, there exists ana ∈ V2 such that x = Ψ(a). By (4.43), the classical Łojasiewicz inequality (4.42)and (4.40),C4c‖M (x)‖0,α ≥ c| f ′(Π0a)|≥ | f (Π0a)− f (0)|1−θ= |E (Ψ(Π0a))−E (Ψ(a))+E (Ψ(a))−E (xc)|1−θ≥ |E (x)−E (xc)|1−θ −C3‖M (x)‖2(1−θ)0,α .(4.44)Since 2(1−θ)≥ 1, (4.34) is established for some C2 > 0 and for all x ∈W0.4.1.3 Proof of Theorem 1.2.1The following lemma is first proved in [17] (Lemma 7.10 therein) when Σ isan n-dimensional self-shrinking embedded hypersurface in Rn+1 with polynomialgrowth. Since a branched conformal immersion is immersed away from finitelymany points, the exact same proof holds for compact branched conformally im-mersed self-shrinkers in Rm,m ≥ 3. For the reader’s convenience, we sketch theproof of Lemma 4.1.5 in the appendix. Note that the F -functional (2.8) and theentropy (2.13) are also defined for branched immersions of compact surfaces.Lemma 4.1.5. Let F : Σ→ Rm,m ≥ 3, be a compact branched conformally im-mersed self-shrinking surface. Then the entropy λ defined in (2.13) is maximizedat (x0, t0) = (0,1). That is,λ (F) =14pi∫Σe−|F |24 dµ. (4.45)56Note that if (F,τ) is a critical point of E , then F is a branched conformally im-mersed self-shrinking surface. Conformality of F then implies |DF |2τdµτ = 2dµ ,where dµ is the area element of the metric induced by F away from the branchpoints. Together with (4.3) and Lemma 4.1.5,E (F,τ) =∫Te−|F |24 dµ = 4piλ (F). (4.46)Now we proceed to prove Theorem 1.2.1.Proof. Assume the theorem is false. Then there is a sequence {Fn} ∈ XΛ withλ (Fi) 6= λ (Fj) for all i 6= j. Let gn = F∗n 〈·, ·〉 and let gτn be the Riemannian metricon T which is of the form (4.4) and is conformal to gn. By Theorem 1.1.3, there isF ∈ XC and τ ∈H so that Fn converges smoothly to F and τn→ τ . Thus‖(Fn,τn)− (F,τ)‖2,α → 0 as n→ ∞.From Proposition 4.1.2 and (4.46) and by setting xc = (F,τ) in (4.34), we haveλ (Fi) = λ (F) for all i large enough, sinceM (Fn,τn) = 0 for all n. That leads to acontradiction. Thus the theorem is proved.4.2 Piecewise Lagrangian mean curvature flowIn this section, we extend the definition of the piecewise mean curvature flow in[17] to Lagrangian mean curvature flow for tori in R4 and construct a piecewiseLagrangian mean curvature flow for a Lagrangian immersed torus F : T→ R4.Definition 4.2.1. Let F : L→R4 be a Lagrangian immersion, where L is a compactsurface. A piecewise Lagrangian mean curvature flow with initial condition F is afinite collection of smooth Lagrangian mean curvature flowsF it : L→ R4defined on [ti, ti+1], i = 0,1, · · · ,k− 1, where 0 = t0 < t1 < · · · < tk−1 < tk < ∞ sothat:1. F00 = F ,572. µ(F i+1ti+1 ) = µ(Fiti+1),3. λ (F i+1ti+1 )< λ (Fiti+1),4. there is δ > 0 such that‖F iti+1−F i+1ti+1 ‖C0 ≤ δ√µ(F iti+1) (4.47)for i = 0,1,2, · · · ,k−2.Remark 2. Note that if k = 1, the piecewise mean curvature flow is just the usualsmooth mean curvature flow. The above definition is interesting only if we cancharacterize the behavior of the flow when t→ tk.Let {Ft : L→ R4} be a smooth mean curvature flow defined on [t0,T0), whereT0 < ∞ and L is a closed surface. Assume that a so-called type I singularity devel-ops at T0, which means supFt(L) ‖At‖→∞ as t→ T0 and there is a positive constantC so thatmaxFt(L)|At |2 ≤ C√T0− t (4.48)for all t < T0. Let tn→ T0 and qn ∈ Ftn(L) where maxFn(L) |Atn | is attained, and sup-pose qn→ q∈R4. Consider the type I rescaling, which is the family of immersionsF˜(·,s), where − logT0 ≤ s < ∞ andF˜(·,s) = 1√(T0− t)(Ft(x)−q), s(t) =− log(T0− t). (4.49)For any sequence s j→ ∞, a subsequence of {F˜(·,s j)} converges locally smoothlyto a self-shrinking immersion F : Σ→ R4 ([29]). In this case, we say that the typeI singularity can be modelled by F . It is not known whether F is unique: If wechoose another sequence s˜k, {F˜(·, s˜k)} might converge to a different self-shrinker.Now we prove Theorem 1.2.2.Proof. Let F : T→ R4 be a Lagrangian immersion. By [51], there is a uniquesmooth Lagrangian mean curvature flow {Ft} which is defined on a maximal timeinterval [0,T0), where T0 < ∞ as T is compact.58If the singularity at T0 is not a type I singularity that can be modelled by a com-pact self-shrinker with area no larger than Λ, then we set k = 0 and no perturbationis performed.Otherwise, the singularity at T0 is of type I and it can be modelled by a com-pact self-shrinker with area no larger than Λ. In this case, the inequality (4.48)is satisfied at a point q ∈ R4 at time T0 for some positive constant C and for allt ∈ [0,T0), and there is a sequence s j→ ∞ such that F˜(·,s j) as in (4.49) convergeslocally smoothly to a compact self-shrinker F with area no bigger than Λ. To beprecise about the convergence, we recall that Lemma 3.3, Corollary 3.2 and Propo-sition 2.3 in [29] hold for any codimension, and they guarantee that all F˜(·,s j)touch a fixed bounded region, the areas inside a ball B(R) are bounded by C(R)and the second fundamental forms and their derivatives of any order are bounded.Therefore, all the conditions in Theorem 1.3 in [6] are satisfied for the sequence{F˜(·,s j)}, and the theorem asserts: by passing to a subsequence if necessary, thereis a surface Σ and an immersion F : Σ→ R4 and a sequence of diffeomorphismsϕ j : U j→ F˜(·,s j)−1(B j)⊂ T,where B j is the ball of radius j in R4 centered at the origin, U j ⊂ Σ are open setswith U j ⊂⊂U j+1 and Σ=⋃ j U j, such that‖F˜(·,s j)◦ϕ j−F‖C0(U j)→ 0and F˜(·,s j)◦ϕ converges to F locally smoothly. In our situation, we have assumedthat Σ is compact (as we are dealing with singularity that can be modelled by com-pact shrinkers). Hence Σ=Uk for all k large and thus ϕk are diffeomorphisms fromΣ to T, since the torus is connected. To simplify notations, we write Σ = T. Thediffeomorphisms ϕ j : T→ T have the property that‖F˜(·,s j)◦ϕ j−F‖Ck(T)→ 0 (4.50)for all k = 0,1,2, · · · . Since each {Ft} is Lagrangian, the sequence of blowupsF˜(·,s j) are also Lagrangian for all j. The above convergence implies that F isLagrangian, hence, F ∈ XΛ.59Since the entropy λ (2.13) is translation and scaling invariant,λ (F˜(·,s(t))) = λ (Ft). (4.51)Furthermore, by the definition ofFx0,t0 in (2.8), we seeλ (F˜(·,s j)◦ϕ j) = λ (F˜(·,s j)). (4.52)Since F0,1 (see (2.8)) is continuous with respect to the C1-topology, there is asequence d j of positive numbers so that d j→ 0 as j→ ∞ andF0,1(F˜(·,s j)◦ϕ j)≥F0,1(F)−d j.By definition of λ and Lemma 4.1.5, since F is a self-shrinker, from the above wehaveλ (F˜(·,s j)◦ϕ j)≥ λ (F)−d j. (4.53)As λ is non-increasing along the mean curvature flow, λ (F˜(·,s j)) is non-increasingin j by (4.51). Together with (4.52) and (4.53), we concludeλ (F˜(·,s j))≥ limj→∞λ (F˜(·,s j))≥ λ (F).Fix δ > 0. Letδ1 =δ√µ(F)6, δ2 = min{12,δ1‖F‖C0 +δ1}.Using (4.50), for all k ≥ 1, there is j0 so that‖F˜(·,s j0)◦ϕ j0−F‖Ck < δ1, (4.54)and ∣∣∣∣∣µ(F˜(·,s j0))µ(F) −1∣∣∣∣∣≤ δ2. (4.55)By Theorem 2.3.1, F is Lagrangian F -unstable. Then by Theorem 2.3.2, there is60a Lagrangian immersion F̂ : T→ R4 which satisfies‖F̂−F‖C2 < δ1, (4.56)∣∣∣∣µ(F)µ(F̂) −1∣∣∣∣≤ δ2 (4.57)andλ (F̂)< λ (F). (4.58)Now we define the first part of the piecewise Lagrangian mean curvature flow:(i) The first piece of Lagrangian mean curvature flow is just F0t := Ft , wheret ∈ [0, t1] and t1 < T0 is such that s(t1) = s j0 .(ii) Define the first perturbation F1t1 at time t1 asF1t1 =√T0− t1(κF̂)◦ϕ−1j0 +q. (4.59)where the dilation factorκ =√µ(F˜(·,s j0)µ(F̂).The constant κ is chosen so thatµ(κF̂) = µ(F˜(·,s j0)). (4.60)We check now that (2)-(4) in definition 4.2.1 are satisfied with i = 0. First notethat (2) follows from (4.60) and the definition of F0t1 and F1t1 . To prove (3), since theentropy (2.13) is scaling and translation invariant, using λ (F)> λ (F̂) we obtainλ (F0t1 ) = λ (F˜(·,s j0)≥ λ (F)> λ (F̂) = λ (F1t1 ).Thus (3) is also shown. Lastly, we show that (4.47) is satisfied with i = 0. From(4.49) and (4.59), we have‖F0t1 −F1t1‖C0 =√T0− t1‖F˜(·,s j0)◦ϕ j0−κF̂‖C0 .61Note that (4.55) and (4.57) imply|κ−1| ≤ B. (4.61)Together with (4.56), (4.54), the definition of δ2, we have‖F˜(·,s j0)◦ϕ j0−κF̂‖C0 ≤ ‖F˜(·,s j0)◦ϕ j0−F‖C0 +‖F− F̂‖C0 +‖(1−κ)F̂‖C0≤ 2δ1+δ2(δ1+‖F˜‖C0)≤ 3δ1,where we used the simple estimate‖F̂‖C0 ≤ ‖F̂−F‖C0 +‖F‖C0 .Thus we have‖F0t1 −F1t1‖C0 < 3δ1√T0− t1= 3δ1√µ(F0t1 )µ(F˜(·,s j0))≤ 12δ√µ(F0t1 )√µ(F)µ(F˜(·,s j0))≤ δ√µ(F0t1 ),where in the last step we used δ2 ≤ 12 . Thus (4.47) is shown and this finishes theconstruction of the first piece of the piecewise Lagrangian mean curvature flow.Using F1t1 as initial condition, there is another family {Ft : t ∈ [t1,T1)} of smoothLagrangian mean curvature flow with Ft1 = F1t1 . Again, if the condition in Theorem1.2.2 is satisfied at the singular time T1 (that is, the singularity at T1 is not of type Iwhich can be modelled by a compact self-shrinker of area ≤ Λ), then we set k = 1,t2 = T1 and we are done. If not, we carry out exactly the same procedure as above.Thus we have a Lagrangian self-shrinking torus F1 ∈ XΛ, some time t2 < T1 andanother Lagrangian immersion F2t2 so thatλ (F1t2 )≥ λ (F1)> λ (F2t2 ),62µ(F1t2 ) = µ(F2t2 )and‖F1t2 −F2t2‖C0 < δ√µ(F1t2 ).Then, again, we apply the smooth Lagrangian mean curvature flow to F2t2 . Note thatthe above procedure must stop: Indeed, by Theorem 1.2.1, the image of λ :XΛ→Ris finite. Moreover, from the above construction, each perturbation is chosen sothat the entropy value is strictly less then one of the element in λ (XΛ). Since λis non-increasing along the usual mean curvature flow, the above procedure mustterminate after k steps for some k ≤ |λ (XΛ)|. This implies that at tk, the piecewiseLagrangian mean curvature flow do not encounter a type I singularity which canbe modelled by a compact self-shrinker with area less than or equals to Λ.To prove the last statement of Theorem 1.2.2, recall that the Maslov class of aLagrangian immersion is given by 2[H]∈H1(T,Z), where H is the mean curvatureform and [H] is an integral class asH = dθ , (4.62)where θ : T→ S1 is the Lagrangian angle of the immersion F : T→ R4 [28].When {Ft} is a smooth Lagrangian mean curvature flow, [Ht ] is invariant as [Ht ]is an integral class and Ht is smooth in t. This fact can also be checked usingthe evolution of H under the Lagrangian mean curvature flow, see Theorem 2.9 in[51]. From (4.62) it is also clear that the Maslov class is invariant under translationand scaling of the immersion. Thus when there is a type I singularity and F :T→ R4 is a compact Lagrangian self-shrinker which models the singularity, then[HF ] = [Ht ]. Lastly, we recall that in Theorem 2.3.2 the perturbation F̂ is definedusing a closed 1-form on T. Hence we also have [HF ] = [HF1t1 ]. Thus the Maslovclass is preserved when we perturb the Lagrangian immersion in constructing thepiecewise Lagrangian mean curvature flow. This completes the proof of Theorem1.2.2.634.2.1 Generalization to Lagrangian immersion of higher genussurfacesTheorem 1.2.2 can be extended to genus g > 1 if we impose further assumptionson the singularity. Let c1,c2 > 0 and consider the set Ximmg,c1,c2 of all Lagrangian self-shrinking immersions F :Σg→R4 with area≤ c1 and the second fundamental formsatisfying maxF(Σg) |A| ≤ c2, where Σg is a closed orientable surface of genus g withg> 1. Using (2.2), there are constants C(k,c1,c2)> 0 that depend on c1,c2,k, suchthatmaxF(Σg)|∇kA| ≤C(k,c1,c2)for all F ∈ Ximmg,c1,c2 . Thus we can apply Theorem 1.3 in [6] to conclude that Ximmg,c1,c2is compact in the C2-topology, in particular, all sequential limits are unbranched.Unbranchedness of any limiting surface guarantees existence of nearby Lagrangianimmersions by the Lagrangian neighbourhood theorem. By Theorem 2.3.1 andTheorem 2.3.2 again, the Lagrangian self-shrinkers in Ximmg,c1,c2 are Lagrangian en-tropy unstable. It follows that all F ∈ Ximmg,c1,c2 are Lagrangian entropy unstable.With these facts, the proof of the following proposition is identical to that of Corol-lary 8.4 in [17] and is omitted here.Proposition 4.2.1. Let δ > 0. Then there is a positive constant c depending onlyon δ such that for any Lagrangian self-shrinker F ∈Ximmg,c1,c2 , there is a Lagrangianimmersion F̂ : Σg→ R4 so that ‖F̂−F‖C0 < δ√µ(F) and λ (F̂)< λ (F)− c.Remark 3. For genus > 1, without assuming uniform boundedness of the sec-ond fundamental forms, the compactness Theorem 1.1.2 is not enough to concludeProposition 4.2.1 due to the assumption on the conformal structures there.Using Proposition 4.2.1, we can define a piecewise Lagrangian mean curvatureflow for a Lagrangian immersion F : Σg→R4, as we did in the genus 1 case. Aftereach perturbation, the entropy decreases by a fixed amount c > 0 (Note that this cmight depend on δ ). Since the entropy is always is positive number, we concludethat the process must terminate in finite time and we have the followingTheorem 4.2.1. Let F : Σg→ R4 be a Lagrangian immersion and δ > 0 be given.Then there exists a piecewise Lagrangian mean curvature flow {F it : i= 0,1, · · · ,k−641} with initial condition F, such that the singularity at time tk is not a type I sin-gularity which can be modelled by a self-shrinker in Ximmg,c1,c2 . Moreover, we havethe estimates ‖F iti −F i+1ti ‖C0 < δ√µ(F iti) and the Maslov class of each immersionis invariant along the flow.4.3 Appendix4.3.1 Proof of Lemma 4.1.5Let Σ be a compact surface without boundary and let F : Σ→ R4 be a branchedconformal self-shrinker. Define the operatorLs byLsu = ∆u− 12ts 〈(x− xs)>,∇u〉= e |x−xs |24ts div(e−|x−xs |24ts ∇u). (4.63)Here (xs, ts) ∈R4×R>0, ∇, div and ∆ are taken with respect to the pullback metricF∗〈·, ·〉 and u,v are functions on R4. Note that Ls is defined away from the set ofbranch points B. As in [17], we use the square bracket [·]s to denote[ f ]s =14pits∫Σf e−|x−xs |24ts dµ (4.64)Lemma 4.3.1. We have[uLsv]s =−[〈∇u,∇v〉]s. (4.65)Proof. Let B = {x1, · · · ,xn}. Let ε > 0 be small and Bi(ε) be an ε-ball in Σ withcenter xi, so that Bi(ε)∩B j(ε) = /0 if i 6= j. Then65[uL v]s =14pits∫Σudiv(e−|x−xs |24ts ∇v)dµ= limε→014pits∫Σ\⋃Bi(ε) udiv(e−|x−xs |24ts ∇v)dµ= limε→014pits(∑i∫∂Bi(ε)u〈∇v,ni〉e−|x−xs|24ts dl−∫Σ\⋃Bi(ε)〈∇u,∇v〉e− |x−xs|24ts dµ)=−[〈∇u,∇v〉]s(4.66)where ni is the unit outward normal along ∂Bi(ε).In particular, we have[uLsv]s =−[〈∇u,∇v〉]s = [vLsu]s. (4.67)Using (4.67), exactly the same argument in [17], pp. 786-788, shows that for ally∈R4 and a∈R if we set (xs, ts) = (sy,1+as2) and g(s) =Fxs,ts(F) then g′(s)≤ 0for all s > 0 with 1+ as2 > 0. Thus Fy,t(F) ≤F0,1(F) for all (y, t) ∈ R4×R>0and thus Lemma 4.1.5 is proved.4.3.2 Analyticity of E andMNext we show that both E and M defined in (4.3) and (4.5) are analytic. For thedefinition of continuous symmetric n-linear form and analytic function betweenBanach spaces, please refer to Chapter 4 in [21]. First we haveLemma 4.3.2. Let X, Y Z be Banach spaces, U , V are open in X, Y respectively,and f :U → R, g : V → Z are analytic at x0 ∈U , y0 ∈ V respectively. Then thefunctionh :U ×V → Z, h(x,y) = f (x)g(y)is analytic at (x0,y0).66Proof. Since f ,g are analytic at x0,y0 respectively, thenf (x0+h) = f (x0)+∞∑i=1Ai(hi), g(y0+ k) = g(y0)+∞∑j=1B j(k j) (4.68)for all ‖h‖X < ε1 and ‖k‖Y < ε2, and Ai,B j are continuous multi-linear forms suchthat ∞∑i=1‖Ai‖ε i1 <+∞ and∞∑j=1‖B j‖ε j2 <+∞. (4.69)The absolute convergence of (4.68) implies thath(x0+h,y0+ k) = f (x0)g(y0)+∞∑n=1Cn(h,k), (4.70)for all (h,k) such that ‖h‖X < ε1,‖k‖Y < ε2, whereCn(h,k) =n∑i=0Ai(hi)Bn−i(kn−i). (4.71)Let ε = 12 min{ε1,ε2}. Then by definition of ‖Cn‖ and ε , one has‖Cn‖εn = sup‖h‖+‖k‖=ε‖Cn(h,k)‖Z≤n∑i=0(‖Ai‖ε i1)(‖Bn−i‖εn−i2 ) .Thus∞∑n=1‖Cn‖εn ≤(∞∑i=1‖Ai‖ε i1)(∞∑j=1‖B j‖ε j2)<+∞by (4.69). Hence h is also analytic at (x0,y0).Proposition 4.3.1. The mapping E :U → R in (4.3) is analytic.Proof. Using (4.4), we have2E (u,τ) =(τ21/τ2+ τ2)L11(u)− (2τ1/τ2)L12(u)+L22(u), (4.72)67whereLi j(u) =∫TDiu ·D jue−|u|24 dxdy. (4.73)Since τ 7→ (τ21/τ2)+ τ2 and τ 7→ τ1/τ2 are analytic, by Lemma 4.3.2, it sufficesto check Li j : C2,α → R is analytic. But this is obvious, using the power seriesexpansion of e−|u|24 .Proposition 4.3.2. The mappingM :U 7→ C 0,α in (4.5) is analytic.Proof. It suffices to show that both components in (4.7) are analytic. The secondcomponent (u,τ) 7→ ∇E uτ is analytic since E is analytic by Proposition 4.3.1, herewe recall that ∇E uτ is the gradient of E (u,τ) at τ . Note that the first component canbe written as(u,τ) 7→ −gi jτ(Di ju− (u ·D ju)Diu+ 14(Diu ·D ju)u)(4.74)Since τ 7→ gi jτ is analytic, the mapping in (4.74) is also analytic by Lemma 4.3.2.68Chapter 5Parabolic Omori-Yau maximumprinciple for mean curvature flowIn this chapter, we prove a parabolic Omori-Yau maximum principle for meancurvature flow and provide some applications. The main reference is [38].5.1 Proof of the parabolic Omori-Yau maximumprincipleWe recall the definition of `-sectional curvature in [35]. Let MN be an N-dimensionalRiemannian manifold. Let p ∈M, 1 ≤ ` ≤ N− 1. Consider a pair {w,V}, wherew ∈ TpM and V ⊂ TpM is a `-dimensional subspace so that w is perpendicular toV .Definition 5.1.1. The `-sectional curvature of {w,V} is given byK`M(w,V ) =`∑i=1〈R(w,ei)w,ei〉, (5.1)where R is the Riemann curvature tensor on M and {e1, · · · ,e`} is any orthonormalbasis of V .We say that M has `-sectional curvature bounded from below by a constant C69ifK`M(w,V )≥ `Cfor all pairs {w,V} at any point p ∈ M. In [35], the authors prove the followingcomparison theorem for the distance function r on manifolds with lower bound on`-sectional curvatures.Theorem 5.1.1. [Theorem 1.2 in [35]] Assume that M has `-sectional curvaturebounded from below by −C for some C > 0. Let p ∈M and r(x) = dg(x, p). If x isnot in the cut locus of p and V ⊂ TxM is perpendicular to ∇r(x), then`∑i=1∇2r(ei,ei)≤ `√C coth(√Cr), (5.2)where {e1, · · · ,e`} is any orthonormal basis of V .Now we prove Theorem 1.3.2. We recall that F is assumed to be proper, and usatisfies condition (1)-(3) in the statement of Theorem 1.3.2.Proof of Theorem 1.3.2. Adding a constant to u if necessary, we assumesupx∈Mu(x,0) = 0.By condition (1) in Theorem 1.3.2, we have u(y,s) > 0 for some (y,s). Notethat s > 0. Let y0 ∈ M and r(y) = dg¯(y,y0) be the distance to y0 in M. Letρ(x, t) = r(F(x, t)). Note that u(y,s)− ερ(y,s)2 > 0 whenever ε is small. Let(x¯i,si) be a sequence so that u(x¯i,si)→ supu ∈ (0,∞]. Let {εi} be a sequence in(0,ε) converging to 0 which satisfiesεiρ(x¯i,si)2 ≤ 1i , i = 1,2, · · · . (5.3)Defineui(x, t) = u(x, t)− εiρ(x, t)2.Note that ui(y,s)> 0 and ui(·,0)≤ 0. Using condition (3) in Theorem 1.3.2, thereis R > 0 so that ui(x, t)≤ 0 when F(x, t) /∈ BR(y0), the closed ball in M centered aty0 with radius R. Since M is complete, BR(y0) is a compact subset. Furthermore,70F is proper and thus ui attains a maximum at some (xi, ti) ∈M× (0,T ]. From thechoice of (x¯i,si) and εi in (5.3),u(xi, ti)≥ ui(xi, ti)≥ ui(x¯i,si)≥ u(x¯i,si)− 1i .Thus we haveu(xi, ti)→ supu.Now we consider the derivatives of u at (xi, ti). If F(xi, ti) is not in the cut locus ofy0, then ρ is differentiable at (xi, ti). Then so is ui and we have∇Mti ui = 0 and(∂∂ t−∆Mti)ui ≥ 0 at (xi, ti). (5.4)(The inequality holds since ti > 0). The first equality implies∇Mti u = εi∇Mtiρ2 = 2εiρ(∇r)> (5.5)at (xi, ti), where (·)> denotes the projection onto TxiMti . Let {e1, · · · ,en} be anyorthonormal basis at TxiMti with respect to gti . Then∆Mtiρ2 = 2n∑i=1|∇Mtiei r|2+2ρn∑i=1∇2r(ei,ei)+2ρ g¯(∇r, ~H). (5.6)Next we use the lower bound on (n− 1)-sectional curvature of M to obtain thefollowing lemma.Lemma 5.1.1. There is C1 =C1(n,C)> 0 so thatn∑i=1∇2r(ei,ei)≤C1. (5.7)Proof of lemma. : We consider two cases. First, if γ ′ is perpendicular to TxiMti ,writen∑i=1∇2r(ei,ei) =1n−1n∑j=1∑i 6= j∇2r(ei,ei).Since M has (n− 1)-sectional curvature bounded from below by −C, we apply71Theorem 5.1.1 to the plane V spanned by {e1, · · · ,en}\{ei} for each i. Thusn∑i=1∇2r(ei,ei)≤ nn−1n−1∑j=1√C coth(√Cρ)= n√C coth(√Cρ).(5.8)Second, if γ ′ is not perpendicular to TxiMti , since the right hand side of (5.7) isindependent of the orthonormal basis chosen, we can assume that e1 is parallel tothe projection of γ ′ onto TxiMti . Writee1 = e⊥1 +aγ′,where e⊥1 lies in the orthogonal complement of γ ′ and a = 〈e1,γ ′〉. By a directcalculation,∇2r(e1,e1) = (∇e1∇r)(e1)= e1〈γ ′,e1〉−〈γ ′,∇e1e1〉= 〈∇e1γ ′,e1〉= 〈∇e⊥1 +aγ ′γ′,e⊥1 +aγ′〉= 〈∇e⊥1 γ′,e⊥1 〉+a〈∇e⊥1 γ′,γ ′〉= ∇2r(e⊥1 ,e⊥1 ).(5.9)We further split into two situations. If e⊥1 = 0, then the above shows ∇2r(e1,e1) =0. Using Theorem 5.1.1 we concluden∑i=1∇2r(ei,ei) =n∑i=2∇2r(ei,ei)≤ (n−1)√C coth(√Cρ)(5.10)If e⊥1 6= 0, write b= ‖e⊥1 ‖ and f1 = b−1e⊥1 . Then { f1,e2, · · · ,en} is an orthonor-72mal basis of an n-dimensional plane in TF(xi,ti)M orthogonal to γ′. Using (5.9),n∑i=1∇2r(ei,ei) = ∇2r(e⊥1 ,e⊥1 )+n∑i=2∇2r(ei,ei)= b2∇2r( f1, f1)+n∑i=2∇2r(ei,ei)= b2(∇2r( f1, f1)+n∑i=2∇2r(ei,ei))+(1−b2)n∑i=2∇2r(ei,ei).Now we apply Theorem 5.1.1 again (note that the first term can be dealt with as in(5.8))n∑i=1∇2r(ei,ei)≤ b2n√C coth(√Cρ)+(1−b2)(n−1)√C coth(√Cρ)≤ n√C coth(√Cρ).(5.11)Summarizing (5.8), (5.10) and (5.11), we haven∑i=1∇2r(ei,ei)≤ n√C coth(√Cρ)≤C1for some C1 =C1(n,C)> 0. Thus the lemma is proved.Using Lemma 5.1.1, (5.6) and ∂ρ∂ t2= 2ρ g¯(∇r, ~H),(∂∂ t−∆Mti)ρ2 =−2n∑i=1|∇Mtiei r|2−2ρn∑i=1∇2r(ei,ei)≥−2n−2C1ρ(5.12)(5.5) and (5.12) imply that at (xi, ti) we have respectively|∇u| ≤ 2εiρ (5.13)and (∂∂ t−∆Mti)u≥−2εi(n+C1ρ). (5.14)73Noteu(xi, ti)− εiρ(xi, ti)2 = ui(xi, ti)≥ ui(y,s)> 0.This impliesρ(xi, ti)2 ≤ u(xi, ti)ε−1i .Using the sub-linear growth condition (3) of u and Young’s inequality, we haveρ(xi, ti)2 ≤ Bε−1i +Bε−1i ρ(xi, ti)α≤ Bε−1i +12ρ(xi, ti)2+12(Bε−1i )22−α .Thus we getρ(xi, ti)εi ≤√2B√εi+B12−α ε1−α2−αi .Together with (5.13), (5.14) and that εi→ 0,|∇u|(xi, ti)→ 0, liminfi→∞(∂∂ t−∆Mti)u(xi, ti)≥ 0.This proves the theorem if ρ is smooth at (xi, ti) for all i. When ρ is not differen-tiable at some (xi, ti), one applies the Calabi’s trick by considering rε(y) = dg¯(y,yε)instead of r, where yε is a point closed to y0. The method is standard and thus isskipped.Remark 4. Condition (1) in the above theorem is used solely to exclude the casethat ui is maximized at (xi,0) for some xi ∈ M. The condition can be dropped ifthat does not happen (see the proof of Theorem 1.3.4).5.2 Preservation of Gauss imageIn this section we assume that F0 : Mn → Rn+m is a proper immersion. Let F :M× [0,T ]→Rn+m be a mean curvature flow starting at F0. We further assume thatthe second fundamental form are uniformly bounded: there is C0 > 0 so that‖A(x, t)‖ ≤C0, for all (x, t) ∈M× [0,T ]. (5.15)Lemma 5.2.1. The mapping F is proper.74Proof. Let B0(r) be the closed ball in Rn+m centered at the origin with radius r.Then by (2.1) and (5.15) we have|F(x, t)−F(x,0)|=∣∣∣∣∫ t0 ∂F∂ s (x,s)ds∣∣∣∣=∣∣∣∣∫ t0 ~H(x,s)ds∣∣∣∣≤√n∫ t0‖A(x,s)‖ds≤C0√nT.Thus if (x, t) ∈ F−1(B0(r)), then x is in F−10 (B0(r +C0√nT )). Let (xn, tn) ∈F−1(B0(r)). Since F0 is proper, a subsequence of {xn} converges to x ∈M. Since[0,T ] is compact, a subsequence of (xn, tn) converges to (x, t), which must be inF−1(B0(r)) since F is continuous. As r > 0 is arbitrary, F is proper.In particular, the parabolic Omori-Yau maximum principle (Theorem 1.3.2)can be applied in this case.Let G(n,m) be the real Grassmanians of n-planes in Rn+m and letγ : M× [0,T ]→ G(n,m), x 7→ F∗TxM (5.16)be the Gauss map of F .Now we prove Theorem 1.3.3, which is a generalization of a Theorem of Wang[53] to the noncompact situation with bounded second fundamental form.Proof of Theorem 1.3.3. Let d : G(n,m)→ R be the distance to Σ. That is d(`) =infL∈Σ d(L, `). Since γ(·,0) ⊂ Σ, we have d ◦ γ = 0 when t = 0. Using chain ruleand (5.15), as dγ = A,d(γ(x, t)) = d(γ(x, t))−d(γ(x,0)) =∫ t0∇d ◦dγ(x,s)ds≤ tC0.Since Σ⊂ G(n,m) is compact, there is ε0 > 0 so that the open setV = {` ∈ G(n,m) : d(`,Σ)<√ε0}75lies in a small tubular neighborhood of Σ and the function d2 is smooth on thisneighborhood. Let T ′ = ε0/2C0. Then the image of f := d2 ◦ γ lies in this tubularneighborhood if t ∈ [0,T ′] and f is a smooth function on M× [0,T ′].The calculation in [53] shows that(∂∂ t−∆)f ≤C|At |2 f , (5.17)where C > 0 depends on ε0 and Σ. Together with (5.15) this shows that(∂∂ t−∆)f ≤C1 ffor some positive constant C1.Let g = e−(C1+1)t f . Then g is bounded, nonnegative and g(·,0) ≡ 0. On theother hand,(∂∂ t−∆)g =−(C1+1)g+ e−(C1+1)t(∂∂ t−∆)f ≤−g. (5.18)If g is positive at some point, Theorem 1.3.2 implies the existence of a sequence(xi, ti) so thatg(xi, ti)→ supg, limsupi→∞(∂∂ t−∆)g(xi, ti)≥ 0.Take i→ ∞ in (5.18) gives 0 ≤ −supg, which contradicts that g is positive some-where. Thus g and so f is identically zero. This is the same as saying thatγ(x, t) ∈ Σ for all (x, t) ∈ [0,T ′]. Note that T ′ depends only on C0, so we canrepeat the same argument finitely many time to conclude that γ(x, t) ∈ Σ for all(x, t) ∈M× [0,T ].Proof of Corollary 1.3.1. An immersion is Lagrangian if and only if its Gauss maphas image in the Lagrangian Grassmanians LG(n), which is a totally geodesic sub-manifold of G(n,n). The Corollary follows immediately from Theorem 1.3.3.Remark 5. Various forms of Corollary 1.3.1 are known to the experts. In [40], theauthor comments that the argument used in [51] can be generalized to the complete76noncompact case, if one assumes the following volume growth condition:Vol(L0∩BR(0))≤C0Rn, for some C0 > 0.The above condition is needed to apply the non-compact maximum principle in[22].5.3 Omori-Yau maximum principle for self-shrinkersIn this section, we improve Theorem 5 in [8] by using Theorem 1.3.2. The proofis more intuitive in the sense that we use essentially the fact that a self-shrinker isa self-similar solution to the mean curvature flow (possibly after reparametrizationon each time slice).First we recall some facts about self-shrinker. A self-shrinker to the meancurvature flow is an immersion F˜ : Mn→ Rn+m which satisfiesF˜⊥ =−12~H. (5.19)Fix T0 ∈ (−1,0). Let φt : M→M be a family of diffeomorphisms on M so thatφT0 = IdM,∂∂ t(F˜(φt(x)))=12(−t) F˜>(φt(x)), ∀t ∈ [−1,T0]. (5.20)LetF(x, t) =√−tF˜(φt(x)), (x, t) ∈M× [−1,T0]. (5.21)Then F satisfies the MCF equation since by (5.19),∂F∂ t(x, t) =∂∂ t(√−tF˜(φt(x)))=− 12√−t F˜(φt(x))+√−t ∂∂ t(F˜(φt(x)))=− 12√−t F˜(φt(x))+12√−t F˜>(φt(x))=1√−t~HF˜(φt(x))= ~HF(x, t).77Lastly, recall theL operator defined in [17]:L f = ∆ f − 12〈∇ f , F˜>〉. (5.22)We are now ready to prove Theorem 1.3.4:Proof of Theorem 1.3.4. Recall T0 ∈ (−1,0). Let u : M× [−1,T0]→R be given byu(x, t) = f (φt(x)), ∀(x, t) ∈M× [−1,T0]. (5.23)Thenu(x, t)≤C(1+ |F˜(φt(x)|α)≤C(−T0)−α/2|F(x, t)|α .Thus we can apply Theorem 1.3.2 (The condition that u(·,0)≡ 0 in Theorem 1.3.2is used only to exclude the case ti =−1. But sinceui(x, t) = f (φt(x))− εi|√−tF˜(φt(x))|2,in order that ui is maximized at (xi, ti) we must have ti = T0. In particular ti 6=−1).Thus there is a sequence (xi,T0) so thatu(xi,T0)→ supu, |∇MT0 u(xi,T0)| → 0, liminfi→∞(∂∂ t−∆MT0)u(xi,T0)≥ 0.Using φT0 = Id and the definition of u, the first condition givesf (xi)→ sup f . (5.24)Since ∇MT0 = 1√−T0∇M, the second condition gives|∇M f (xi)| → 0. (5.25)Lastly,∂u∂ t(xi,T0) =∂ f∂ t(φt(x))∣∣∣∣t=T0=12(−T0)〈∇ f (xi), F˜>(xi)〉 (5.26)78and∆MT0 u(xi,T0) = ∆MT0 f (xi) =1−T0∆M f (xi).Thus (∂∂ t−∆MT0)u(xi,T0) =1T0L f (xi)and the result follows.Remark 6. Note that the above theorem is stronger than Theorem 5 in [8], wherethey assume that f is bounded above (which corresponds to our case when α = 0).Remark 7. Our growth condition on f is optimal: the function f (x) =√|x|2+1defined on Rn (as a self-shrinker) has linear growth, but the gradient of f∇ f =x√|x|2+1does not tend to 0 as f (x)→ sup f = ∞.Remark 8. 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