Uniqueness of blowups and Lojasiewicz inequalities
aa r X i v : . [ m a t h . DG ] M a y UNIQUENESS OF BLOWUPS AND LOJASIEWICZ INEQUALITIES
TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II
We dedicate this article to Leon Simon in recognition of his fundamentalcontributions to analysis and geometry.
Abstract.
Once one knows that singularities occur, one naturally wonders what the singu-larities are like. For minimal varieties the first answer, already known to Federer-Fleming in1959, is that they weakly resemble cones . For mean curvature flow, by the combined workof Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, thesimple proofs leave open the possibility that a minimal variety or a mean curvature flowlooked at under a microscope will resemble one blowup, but under higher magnification, itmight (as far as anyone knows) resemble a completely different blowup. Whether this everhappens is perhaps the most fundamental question about singularities. It is this long stand-ing open question that we settle here for mean curvature flow at all generic singularities andfor mean convex mean curvature flow at all singularities. Introduction
We show that at each generic singularity of a mean curvature flow the blowup is unique;that is independent of the sequence of rescalings. This settles a major open problem that wasopen even in the case of mean convex hypersurfaces where it was known that all singularitiesare generic. Moreover, it is the first general uniqueness theorem for blowups to a GeometricPDE at a non-compact singularity.Uniqueness of blowups is perhaps the most fundamental question that one can ask aboutsingularities and implies regularity of the singular set; see [CM5].To prove our uniqueness result, we prove two completely new infinite dimensional Lo-jasiewicz type inequalities. Infinite dimensional Lojasiewicz inequalities were pioneered thirtyyears ago by Leon Simon. However, unlike all other infinite dimensional Lojasiewicz in-equalities we know of, ours do not follow from a reduction to the classical finite-dimensionalLojasiewicz inequalities from the 1960s from algebraic geometry, rather we prove our in-equalities directly and do not rely on Lojasiewicz’s arguments or results.It is well-known that to deal with non-compact singularities requires entirely new ideasand techniques as one cannot argue as in Simon’s work, and all the later work that uses hisideas. Partly because of this we expect that the techniques and ideas developed here haveapplications to other flows. Our results hold in all dimensions.
The authors were partially supported by NSF Grants DMS 11040934, DMS 1206827, and NSF FRG grantsDMS 0854774 and DMS 0853501. See Brian White [W4] section “Uniqueness of tangent cone” from which part of this discussion is takenand where one can find more discussion of uniqueness for minimal varieties.
This paper focuses on mean curvature flow (or MCF) of hypersurfaces. This is a non-linearparabolic evolution equation where a hypersurface evolves over time by locally moving inthe direction of steepest descent for the volume element. It has been used and studied inmaterial science for almost a century to model things like cell, grain, and bubble growth .Unlike some of the other earlier papers in material science both von Neumann’s 1952 paperand Mullins 1956 paper had explicit equations. In his paper von Neumann discussed soapfoams whose interface tend to have constant mean curvature whereas Mullins is describingcoarsening in metals, in which interfaces are not generally of constant mean curvature. Partlyas a consequence, Mullins may have been the first to write down the MCF equation in general.Mullins also found some of the basic self-similar solutions like the translating solution nowknown as the Grim Reaper. To be precise, suppose that M t ⊂ R n +1 is a one-parameterfamily of smooth hypersurfaces, then we say that M t flows by the MCF if(0.1) x t = − H n , where H and n are the mean curvature and unit normal, respectively, of M t at the point x .To understand singularities past the first singular time, we need weak solutions of MCF.The weak solutions that we will use are the Brakke flows considered by White in [W3] . Bytheorem 7 . Tangent flows.
By definition, a tangent flow is the limit of a sequence of rescalingsat a singularity, where the convergence is on compact subsets. For instance, a tangentflow to M t at the origin in space-time is the limit of a sequence of rescaled flows δ i M δ i t where δ i →
0. A priori, different sequences δ i could give different tangent flows and thequestion of the uniqueness of the blowup - independent of the sequence - is a major questionin many geometric problems. By a monotonicity formula of Huisken, [H1], and an argumentof Ilmanen and White, [I1], [W3], tangent flows are shrinkers, i.e., self-similar solutions ofMCF that evolve by rescaling. The only generic shrinkers are round cylinders by [CM1].We will say that a singular point is cylindrical if at least one tangent flow is a multiplicityone cylinder S k × R n − k . Our main application of our analytical inequalities is the followingtheorem that shows that tangent flows at generic singularities are unique: Theorem 0.2.
Let M t be a MCF in R n +1 . At each cylindrical singular point the tangentflow is unique. That is, any other tangent flow is also a cylinder with the same R k factorthat points in the same direction. See, e.g., the early work in material science from the 1920s, 1940s, and 1950s of T. Sutoki, [Su], D. Harkerand E. Parker, [HaP], J. Burke, [Bu], P.A. Beck, [Be], J. von Neumann, [N], and W.W. Mullins, [M]. For instance, annealing, in metallurgy and materials science, is a heat treatment that alters a material toincrease its ductility and to make it more workable. It involves heating material above its critical temperature,maintaining a suitable temperature, and then cooling. Annealing can induce ductility, soften material, relieveinternal stresses, refine the structure by making it homogeneous, and improve cold working properties. Thethree stages of the annealing process that proceed as the temperature of the material is increased are:recovery, recrystallization, and grain growth. Grain growth is the increase in size of grains (crystallites) ina material at high temperature. This occurs when recovery and recrystallisation are complete and furtherreduction in the internal energy can only be achieved by reducing the total area of grain boundary [by meancurvature flow]. That is, Brakke flows in the class S ( λ , n, n + 1) defined in section 7 of [W3] for some λ > This is analogous to a tangent cone at a singularity of a minimal variety, cf. [FFl].
This theorem solves a major open problem; see, e.g., page 534 of [W2]. Even in the case ofthe evolution of mean convex hypersurfaces where all singularities are cylindrical, uniquenessof the axis was unknown; see [HS1], [HS2], [W1], [SS], [An] and [HaK]. In recent joint work with Tom Ilmanen, [CIM], we showed that if one tangent flow at asingular point of a MCF is a multiplicity one cylinder, then all are. However, [CIM] left openthe possibility that the direction of the axis (the R k factor) depended on the sequence ofrescalings. Our proof of Theorem 0.2 and, in particular, our first Lojasiewicz type inequality,has its roots in some ideas and inequalities from [CIM] and in fact implicitly use that cylindersare isolated among shrinkers by [CIM].Uniqueness is a key question for the regularity of Geometric PDE’s. Two of the mostprominent early works on uniqueness of tangent cones are Leon Simon’s hugely influentialpaper [Si1] from 1983, where he proves uniqueness for tangent cones of minimal varietieswith smooth cross-section. The other is Allard-Almgren’s 1981 paper [AA], where unique-ness of tangent cones with smooth cross-section is proven under an additional integrabilityassumption on the cross-section; see also [Si2], [Hr], [CM4] for additional references.Our results are the first general uniqueness theorems for tangent flows to a geometricflow at a non-compact singularity. (In fact, not only are the singularities that we deal withhere non-compact but they are also non-integrable; see Section 3.) Some special cases ofuniqueness of tangent flows for MCF were previously analyzed assuming either some sort ofconvexity or that the hypersurface is a surface of rotation; see [H1], [H2], [HS1], [HS2], [W1],[SS], [AAG], section 3 . R n +1 with only cylindrical singularities, the space-time singular setis contained in finitely many compact embedded ( n − n −
2. In particular, if the initial hypersurface ismean convex, then all singularities are generic and the results apply. In fact, in [CM5] weshowed that the entire stratification of the space-time singular set is rectifiable in a verystrong sense; cf., e.g., [Si3], [Si4], [Si5], [BrCoL] and [HrLi].One of the significant difficulties that we overcome in this paper, and sets it apart fromall other work we know of, is that our singularities are noncompact. This causes majoranalytical difficulties and to address them requires entirely new techniques and ideas. Thisis not so much because of the subtleties of analysis on noncompact domains, though this is anissue, but crucially because the evolving hypersurface cannot be written as an entire graphover the singularity no matter how close we get to the singularity. Rather, the geometry ofthe situation dictates that only part of the evolving hypersurface can be written as a graphover a compact piece of the singularity. Lojasiewicz inequalities.
The main technical tools that we prove are two Lojasiewicz–type inequalities. Our results not only give uniqueness of tangent flows but also a definite rate where the rescaled MCFconverges to the relevant cylinder. The distance to the cylinder is decaying to zero at a definite rate overballs whose radii are increasing at a definite rate to infinity. In the end, what comes out of our analysis is that the domain the evolving hypersurface is a graph overis expanding in time and at a definite rate, but this is not all all clear from the outset; see also footnote 3.
TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II
In real algebraic geometry, the Lojasiewicz inequality, [L], named after Stanislaw Lo-jasiewicz, gives an upper bound for the distance from a point to the nearest zero of a givenreal analytic function. Specifically, let f : U → R be a real-analytic function on an openset U in R n , and let Z be the zero locus of f . Assume that Z is not empty. Then for anycompact set K in U , there exist α ≥ C such that, for all x ∈ K inf z ∈ Z | x − z | α ≤ C | f ( x ) | . (0.3)Here α can be large.Lojasiewicz, [L], also proved the following inequality : With the same assumptions on f ,for every p ∈ U , there is a possibly smaller neighborhood W of p and constants β ∈ (0 , C > x ∈ W | f ( x ) − f ( p ) | β ≤ C |∇ x f | . (0.4)Note that this inequality is trivial unless p is a critical point for f .An immediate consequence of (0.4) is that every critical point of f has a neighborhoodwhere every other critical point has the same value. Lojasiewicz inequalities for non-compact hypersurfaces and MCF.
The infinitedimensional Lojasiewicz-type inequalities that we prove are for the F functional on the spaceof hypersurfaces.The F -functional is given by integrating the Gaussian over a hypersurface Σ ⊂ R n +1 .This is also often referred to as the Gaussian surface area and is defined by F (Σ) = (4 π ) − n/ Z Σ e − | x | dµ . (0.5)The entropy λ (Σ) is the supremum of the Gaussian surface areas over all centers and scales.It follows from the first variation formula that the gradient of F is ∇ Σ F ( ψ ) = Z Σ (cid:18) H − h x, n i (cid:19) ψ e − | x | . (0.6)Thus, the critical points of F are shrinkers, i.e., hypersurfaces with H = h x, n i . The mostimportant shrinkers are the generalized cylinders C ; these are the generic ones by [CM1].The space C is the union of C k for k ≥
1, where C k is the space of cylinders S k × R n − k , wherethe S k is centered at 0 and has radius √ k and we allow all possible rotations by SO ( n + 1).A family of hypersurfaces Σ s evolves by the negative gradient flow for the F -functional ifit satisfies the equation ( ∂ s x ) ⊥ = − H n + x ⊥ / . (0.7)This flow is called the rescaled MCF since Σ s is obtained from a MCF M t by setting Σ s = √− t M t , s = − log( − t ), t <
0. By (0.6), critical points for the F -functional or, equivalently,stationary points for the rescaled MCF, are the shrinkers for the MCF that become extinctat the origin in space-time. A rescaled MCF has a unique asymptotic limit if and only if thecorresponding MCF has a unique tangent flow at that singularity. Lojasiewicz called this inequality the gradient inequality. This consequence of (0.4) for the F functional near a cylinder is implied by the rigidity result of [CIM]. We will prove versions of the two Lojasiewicz inequalities for the F functional on a generalhypersurface Σ. Roughly speaking, we will show thatdist(Σ , C ) ≤ C |∇ Σ F | , (0.8) ( F (Σ) − F ( C )) ≤ C |∇ Σ F | . (0.9)Equation (0.8) will correspond to Lojasiewicz’s first inequality whereas (0.9) will correspondto his second inequality. The precise statements of these inequalities will be much morecomplicated than this, but they will be of the same flavor.0.4. First Lojasiewicz with α = 2 implies the second with β = . In this subsectionwe will explain how the second Lojasiewicz inequality for a function f in a neighborhood ofan isolated critical point follows from the first when the first holds for ∇ f and with α = 2.(We will later extend this argument to infinite dimensions.)Suppose that f : R n → R is smooth function with f (0) = 0 and ∇ f (0) = 0; without lossof generality we may assume that at 0 the Hessian is in diagonal form and we will write thecoordinates as x = ( y, z ) where y are the coordinates where the Hessian is nondegenerate.By Taylor’s formula in a small neighborhood of 0, we have that f ( x ) = a i y i + O ( | x | ) . (0.10) f y i ( x ) = a i y i + O ( | x | ) . (0.11) f z i ( x ) = O ( | x | ) . (0.12)It follows from this that the second of the two Lojasiewicz inequalities holds for f and β = provided that | z | ≤ ǫ | y | for some sufficiently small ǫ >
0. Namely, if | z | ≤ ǫ | y | , then C | y | ≤ |∇ x f | and | f ( x ) | ≤ C − | y | (0.13)for some positive constant C and, hence, | f ( x ) | ≤ C |∇ x f | . (0.14)Therefore, we only need to prove the second Lojasiewicz inequality for f in the region | z | ≥ ǫ | y | . We will do this using the first Lojasiewicz inequality for ∇ f . Since 0 is anisolated critical point for f , the first Lojasiewicz inequality for ∇ f gives that |∇ x f | ≥ C | x | . (0.15)By assumption on the region and the Taylor expansion for f , we get that in this region | f ( x ) | ≤ C | y | + C | z | ≤ C | z | ≤ C | x | . (0.16)Combining these two inequalities gives | f ( x ) | ≤ C | x | ≤ |∇ x f | . (0.17)This proves the second Lojasiewicz inequality for f with β = .In Section 4, we extend the above argument to general Banach spaces.Lojasiewicz used his second inequality to show the “Lojasiewicz theorem”: If f : R n → R is an analytic function, x = x ( t ) : [0 , ∞ ) → R n is a curve with x ′ ( t ) = −∇ f and x ( t ) has a TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II limit point x ∞ , then the length of the curve is finite and lim t →∞ x ( t ) = x ∞ . Moreover, x ∞ is a critical point for f .Even in R , it is easy to construct smooth functions where the Lojasiewicz theorem doesnot hold, but instead there are negative gradient flow lines with multiple limits.We will discuss the Lojasiewicz theorem in a slightly more general setting at the end ofthe next subsection after briefly discussing infinite dimensional Lojasiewicz inequalities.0.5. Infinite dimensional Lojasiewicz inequalities and applications.
Infinite dimen-sional versions of Lojasiewicz inequalities were proven in a celebrated work of Leon Simon,[Si1], for the area and related functionals and used, in particular, to prove a fundamentalresult about uniqueness of tangent cones with smooth cross section of minimal surfaces. Si-mon’s proof of the Lojasiewicz inequality is done by reducing the infinite dimensional versionto the classical Lojasiewicz inequality by a Lyapunov-Schmidt reduction argument. Infinitedimensional Lojasiewicz inequalities proven using Lyapunov-Schmidt reduction, as in thework of Simon, have had a profound impact on various areas of analysis and geometry andare usually referred to as Lojasiewicz-Simon inequalities.As already mentioned, we will also prove two infinite dimensional Lojasiewicz inequalitiesand use them to prove uniqueness of blowups for MCF (or, equivalently, convergence ofthe rescaled flow). However, unlike all other infinite dimensional Lojasiewicz inequalitieswe know of, ours do not follow from a reduction to the classical Lojasiewicz inequalities;rather we prove our inequalities directly and do not rely on Lojasiewicz’s arguments orresults. In fact, we prove our infinite dimensional analog of the first Lojasiewicz inequalitydirectly and use this together with an infinite dimensional analog of the argument in theprevious subsection to show our second Lojasiewicz inequality. The reason why we cannotargue as in Simon’s work, and all the later work that make use his ideas, comes from thatour singularities are noncompact. In particular, even near the singularities, the evolvinghypersurface cannot be written as an entire graph over the singularity. Rather, only part ofthe evolving hypersurface can be written as a graph over a compact piece of the singularity.Next we will explain how the second Lojasiewicz inequality is typically used to showuniqueness. Before we do that, observe first that in the second inequality we always work ina small neighborhood of p so that, in particular, | f ( x ) − f ( p ) | ≤ X is a Banach space and f : X → R is a Frechet differentiable function.Let x = x ( t ) be a curve on X parametrized on [0 , ∞ ) whose velocity x ′ = −∇ f . We wouldlike to show that if the second inequality of Lojasiewicz holds for f with a power 1 > β > / x ( t ) has a limit point x ∞ ,then the length of the curve is finite and lim t →∞ x ( t ) = x ∞ . Since x ∞ is a limit point of x ( t )and f is non-increasing along the curve, x ∞ must be a critical point for f .To see that x ( t ) converges to x ∞ , assume that f ( x ∞ ) = 0 and note that if we set f ( t ) = f ( x ( t )), then f ′ = −|∇ f | . Moreover, by the second Lojasiewicz inequality, we get that f ′ ≤ − f β if x ( t ) is sufficiently close to x ∞ . (Assume for simplicity below that x ( t ) stays in asmall neighborhood x ∞ for t sufficiently large so that this inequality holds; the general casefollows with trivial changes.) Then this inequality can be rewritten as ( f − β ) ′ ≥ (2 β − which integrates to f ( t ) ≤ C t − β − . (0.18)We need to show that (0.18) implies that R ∞ |∇ f | ds is finite. This shows that x ( t )converges to x ∞ as t → ∞ . To see that R ∞ |∇ f | ds is finite, observe by the Cauchy-Schwarzinequality that Z ∞ |∇ f | ds = Z ∞ p − f ′ ds ≤ (cid:18) − Z ∞ f ′ s ǫ ds (cid:19) (cid:18)Z ∞ s − − ǫ ds (cid:19) . (0.19)It suffices therefore to show that − Z T f ′ s ǫ ds (0.20)is uniformly bounded. Integrating by parts gives Z T f ′ s ǫ ds = | f s ǫ | T − (1 + ǫ ) Z T f s ǫ ds . (0.21)If we choose ǫ > β , then we see that this is boundedindependent of T and hence R ∞ |∇ f | ds is finite.We will use an extension of this argument where the assumption f β ( t ) ≤ − f ′ ( t ) is replacedby the assumption that f β ( t ) ≤ f ( t − − f ( t + 1); see Lemma 6.22. This assumption isexactly what comes out of our analog for the rescaled MCF of the gradient Lojasiewiczinequality, i.e., out of Theorem 0.26.0.6. The two Lojasiewicz inequalities.
We will now state the two Lojasiewicz-type in-equalities for the F functional on the space of hypersurfaces.Suppose that Σ ⊂ R n +1 is a hypersurface and fix some small ǫ > ℓ and constant C ℓ , we let r ℓ (Σ) be the maximal radius so that • B r ℓ (Σ) ∩ Σ is the graph over a cylinder in C k of a function u with k u k C ,α ≤ ǫ and |∇ ℓ A | ≤ C ℓ .The parameters ℓ and C ℓ will be left free until the proof of the main theorem (Theorem 0.2)and will then be chosen large.In the next theorem, we will use a Gaussian L distance d C ( R ) to the space C k in the ballof radius R . To define this, given Σ k ∈ C k , let w Σ k : R n +1 → R denote the distance to theaxis of Σ k (i.e., to the space of translations that leave Σ k invariant). Then we define d C ( R ) = inf Σ k ∈C k k w Σ k − √ k k L ( B R ) ≡ inf Σ k ∈C k Z B R ∩ Σ k ( w Σ k − √ k ) e − | x | . (0.22)The Gaussian L p norm on the ball B R is k u k pL p ( B R ) = R B R | u | p e − | x | .Given a general hypersurface Σ, it is also convenient to define the function φ by φ = h x, n i − H , (0.23)so that φ is minus the gradient of the functional F . TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II
The main tools that we develop here are the following two analogs for non-compact hy-persurfaces of the well-known Lojasiewicz’s inequalities for analytic functions on R n . Theorem 0.24. (A Lojasiewicz inequality for non-compact hypersurfaces). If Σ ⊂ R n +1 isa hypersurface with λ (Σ) ≤ λ and R ∈ [1 , r ℓ (Σ) − d C ( R ) ≤ C R ρ (cid:26) k φ k b ℓ,n L ( B R ) + e − bℓ,n R (cid:27) , (0.25)where C = C ( n, ℓ, C ℓ , λ ), ρ = ρ ( n ) and b ℓ,n ∈ (0 ,
1) satisfies lim ℓ →∞ b ℓ,n = 1.The theorem bounds the L distance to C k by a power of k φ k L , with an error term thatcomes from a cutoff argument since Σ is non-compact and is not globally a graph of thecylinder. This theorem is essentially sharp. Namely, the estimate (0.25) does not hold forany exponent b ℓ,n larger than one, but Theorem 0.24 lets us take b ℓ,n arbitrarily close to one.We will also see that the above inequality implies the following gradient type Lojasiewiczinequality. This inequality bounds the difference of the F functional near a critical point bytwo terms. The first is essentially a power of ∇ F , while the second (exponentially decaying)term comes from that Σ is not a graph over the entire cylinder. Theorem 0.26. (A gradient Lojasiewicz inequality for non-compact hypersurfaces). If Σ ⊂ R n +1 is a hypersurface with λ (Σ) ≤ λ , β ∈ [0 , R ∈ [1 , r ℓ (Σ) − | F (Σ) − F ( C k ) | ≤ C R ρ (cid:26) k φ k c ℓ,n β β L ( B R ) + e − cℓ,n (3+ β ) R β ) + e − (3+ β )( R − (cid:27) , (0.27)where C = C ( n, ℓ, C ℓ , λ ), ρ = ρ ( n ) and c ℓ,n ∈ (0 ,
1) satisfies lim ℓ →∞ c ℓ,n = 1.When we apply the theorem, the parameters β and ℓ will be chosen to make the exponentgreater than one on the ∇ F term, essentially giving that | F (Σ) − F ( C k ) | is bounded bya power greater than one of |∇ F | . A separate argument will be needed to handle theexponentially decaying error terms.Throughout the paper C will denote a constant that can change from line to line.We will show that when Σ t are flowing by the rescaled MCF, then both terms on the right-hand side of (0.27) are bounded by a power greater than one of k φ k L (the correspondingstatement holds for Theorem 0.24). Thus, we will essentially get the inequalities d C ≤ C |∇ Σ t F | , (0.28) ( F (Σ t ) − F ( C )) ≤ C |∇ Σ t F | . (0.29)These two inequalities can be thought of as analogs for the rescaled MCF of Lojasiewiczinequalities from real algebraic geometry; cf. (0.8) and (0.9).See [CM6] for a survey on Lojasiewicz inequalities and their applications. This is a Lojasiewicz inequality for the gradient of the F functional ( φ is the gradient of F ). This followssince, by [CIM], cylinders are isolated critical points for F and, thus, d C locally measures the distance to thenearest critical point. Cylindrical estimates for a general hypersurface
In this section, we will prove estimates for a general hypersurface Σ ⊂ R n +1 . The mainresults are bounds for ∇ AH when the mean curvature H is positive on a large set.1.1. A general Simons equation.
In this subsection, we will show that the second fun-damental form A of Σ satisfies an elliptic differential equation similar to Simons’ equationfor minimal surfaces. The elliptic operator will be the L operator from [CM1] given by L ≡ L + | A | + 12 ≡ ∆ − ∇ x T + | A | + 12 . (1.1)Namely, we will prove the following proposition: Proposition 1.2. If φ = h x, n i − H , then L A = A + Hess φ + φ A , (1.3)where the tensor A is given in orthonormal frame by ( A ) ij = A ik A kj .Note that φ vanishes precisely when Σ is a shrinker and, in this case, we recover theSimons’ equation for A for shrinkers from [CM1].We will use the following general version of Simons’ equation for the second fundamentalform of a hypersurface: Lemma 1.4.
The second fundamental form A satisfies (cid:0) ∆ + | A | (cid:1) A = − H A − Hess H . (1.5)See, e.g., [CM3] for a proof.The next lemma computes the Hessian of the support function h x, n i . Lemma 1.6.
The Hessian of h x, n i is given byHess h x, n i = −∇ x T A − A − A h x, n i . (1.7) Proof.
Fix a point p ∈ Σ. Let e i be a local orthonormal frame for Σ with ∇ Te i e j = 0 at p forevery i and j . Thus, at p , we have ∇ e i e j = A ij n . (1.8)Finally, using this and ∇ e i n = − A ik e k (which holds at all points), we compute at p Hess h x, n i ( e i , e j ) = h x, n i ij = h x, ∇ e i n i j = − ( A ik h x, e k i ) j = − A ikj h x, e k i − A ik δ jk − A ik h x, A jk n i (1.9) = − ( ∇ x T A ) ( e i , e j ) − A ( e i , e j ) − h x, n i A ( e i , e j ) , where the last equality used the Codazzi equation A ikj = A ijk . (cid:3) Proof of Proposition 1.2.
Since L = L + | A | + and L = ∆ − ∇ x T , Lemma 1.4 gives LA = (cid:0) ∆ + | A | (cid:1) A + 12 A − ∇ x T A = − H A − Hess H + 12 A − ∇ x T A . (1.10)
On the other hand, Lemma 1.6 givesHess φ = 12 Hess h x, n i − Hess H = − Hess H − ∇ x T A − A − A h x, n i , (1.11)so we have L A − Hess φ = A + φ A . (cid:3) An integral bound when the mean curvature is positive.
We will show that thetensor τ = A/H is almost parallel when H is positive and φ is small. This generalizes anestimate from [CIM] in the case where Σ is a shrinker (i.e., φ ≡
0) with
H > f >
0, define a weighted divergence operator div f and drift Laplacian L f bydiv f ( V ) = 1 f e | x | / div Σ (cid:16) f e −| x | / V (cid:17) , (1.12) L f u ≡ div f ( ∇ u ) = L u + h∇ log f, ∇ u i . (1.13)Here u may also be a tensor; in this case the divergence traces only with ∇ . Note that L = L . We recall the quotient rule (see lemma 4 . Lemma 1.14.
Given a tensor τ and a function g with g = 0, then L g τg = g L τ − τ L gg = g L τ − τ L gg . (1.15) Proposition 1.16.
On the set where
H >
0, we have L H AH = Hess φ + φ A H + A (∆ φ + φ | A | ) H , (1.17) L H | A | H = 2 (cid:12)(cid:12)(cid:12)(cid:12) ∇ AH (cid:12)(cid:12)(cid:12)(cid:12) + 2 h Hess φ + φ A , A i H + 2 | A | (∆ φ + φ | A | ) H . (1.18) Proof.
The trace of Proposition 1.2 ( H is minus the trace of A by convention) gives L H = H − ∆ φ − φ | A | , (1.19)where we also used that the trace of A is | A | since A is symmetric. Using the quotient rule(Lemma 1.14) and the equations for LH and LA (from Proposition 1.2) gives L H AH = H LA − A LHH = H ( A + Hess φ + φ A ) − A ( H − ∆ φ − φ | A | ) H = Hess φ + φ A H + A (∆ φ + φ | A | ) H , (1.20)giving the first claim. The second claim follows from the first since | A | H = h AH , AH i and12 L H h AH , AH i = hL H AH , AH i + (cid:12)(cid:12)(cid:12)(cid:12) ∇ AH (cid:12)(cid:12)(cid:12)(cid:12) . (1.21) (cid:3) The next proposition gives exponentially decaying integral bounds for ∇ ( A/H ) when H is positive on a large ball. It will be important that these bounds decay rapidly. Proposition 1.22. If B R ∩ Σ is smooth with
H >
0, then for s ∈ (0 , R ) we have Z B R − s ∩ Σ (cid:12)(cid:12)(cid:12)(cid:12) ∇ AH (cid:12)(cid:12)(cid:12)(cid:12) H e − | x | ≤ s sup B R ∩ Σ | A | Vol( B R ∩ Σ) e − ( R − s )24 + 2 Z B R ∩ Σ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) h Hess φ , A i + | A | H ∆ φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) h A , A i + | A | H (cid:12)(cid:12)(cid:12)(cid:12) | φ | (cid:27) e − | x | . (1.23) Proof.
Set τ = A/H and u = | τ | = | A | /H . It will be convenient within this proof to usesquare brackets [ · ] to denote Gaussian integrals over B R ∩ Σ, i.e. [ f ] = R B R ∩ Σ f e −| x | / .Let ψ be a function with support in B R . Using the divergence theorem, the formula fromProposition 1.16 for L H u , and the absorbing inequality 4 ab ≤ a + 4 b , we get0 = (cid:2) div H (cid:0) ψ ∇ u (cid:1) H ]=[ (cid:0) ψ L H u + 2 ψ h∇ ψ, ∇ u i (cid:1) H (cid:3) = (cid:20)(cid:26) ψ |∇ τ | + 2 ψ (cid:18) h Hess φ + φ A , A i H + | A | (∆ φ + φ | A | ) H (cid:19) + 4 ψ h∇ ψ, τ · ∇ τ i (cid:27) H (cid:21) ≥ (cid:2)(cid:0) ψ |∇ τ | − | τ | |∇ ψ | (cid:1) H (cid:3) + 2 (cid:2) ψ h Hess φ + φ A , A i (cid:3) + 2 (cid:20) ψ | A | (∆ φ + φ | A | ) H (cid:21) , from which we obtain (cid:2) ψ |∇ τ | H (cid:3) ≤ (cid:2) |∇ ψ | | A | (cid:3) − (cid:2) ψ h Hess φ + φA , A i (cid:3) − (cid:20) ψ ∆ φ | A | H + ψ φ | A | H (cid:21) . The proposition follows by choosing ψ ≡ B R − s and going to zero linearly on ∂B R . (cid:3) We record the following corollary:
Corollary 1.24. If B R ∩ Σ is smooth with
H > δ > | A | ≤ C , then there exists C = C ( n, δ, C ) so that for s ∈ (0 , R ) we have Z B R − s ∩ Σ (cid:12)(cid:12)(cid:12)(cid:12) ∇ AH (cid:12)(cid:12)(cid:12)(cid:12) e − | x | ≤ C s Vol( B R ∩ Σ) e − ( R − s )24 + C Z B R ∩ Σ {| Hess φ | + | φ |} e − | x | . (1.25) Remark 1.26.
Corollary 1.24 essentially bounds the distance squared to the space of cylin-ders by k φ k L . This is sharp: it is not possible to get the sharper bound where the powersare the same. This is a general fact when there is a non-integrable kernel. Namely, if weperturb in the direction of the kernel, then φ vanishes quadratically in the distance.The next corollary combines the Gaussian L bound on ∇ τ from Corollary 1.24 withstandard interpolation inequalities to get pointwise bounds on ∇ τ and ∇ τ . Corollary 1.27. If B R ∩ Σ is smooth with
H > δ > | A | + (cid:12)(cid:12) ∇ ℓ +1 A (cid:12)(cid:12) ≤ C , and λ (Σ) ≤ λ ,then there exists C = C ( n, λ , δ, ℓ, C ) so that for | y | + | y | < R −
1, we have (cid:12)(cid:12)(cid:12)(cid:12) ∇ AH (cid:12)(cid:12)(cid:12)(cid:12) ( y ) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ AH (cid:12)(cid:12)(cid:12)(cid:12) ( y ) ≤ C R n (cid:26) e − d ℓ,n ( R − + k φ k dℓ,n L ( B R ) (cid:27) e | y | , (1.28)where the exponent d ℓ,n ∈ (0 ,
1) has lim ℓ →∞ d ℓ,n = 1. Proof.
Set τ = A/H and note that (cid:12)(cid:12) ∇ ℓ +1 τ (cid:12)(cid:12) is bounded by a constant depending on δ , ℓ and C . Define the ball B y and constant δ y by B y = B | y | ( y ) and δ y = Z B y ∩ Σ |∇ τ | . (1.29)Applying Lemma B.1 on B y gives |∇ τ | ( y ) ≤ C ′ n R n δ y + δ a ℓ,n y k∇ ℓ +1 τ k − a ℓ,n L ∞ ( B y ) o ≤ C (cid:8) R n δ y + δ a ℓ,n y (cid:9) , |∇ τ | ( y ) ≤ C ′ n R n +1 δ y + δ b ℓ,n y k∇ ℓ +1 τ k − b ℓ,n L ∞ ( B y ) o ≤ C n R n +1 δ y + δ b ℓ,n y o , where the powers are given by a ℓ,n = ℓ ℓ + n and b ℓ,n = ℓ − ℓ + n , and C = C ( n, δ, ℓ, C ).To get the bound on δ y , observe thatinf B y e − | x | ≥ e − | y | − , (1.30)so that Cauchy-Schwarz gives(1 + | y | ) n e −| y | − δ y ≤ C e −| y | − Z B y ∩ Σ |∇ τ | ≤ C Z B y ∩ Σ |∇ τ | e − | x | ≤ C γ , (1.31)where the last inequality is Corollary 1.24, C = C ( n, λ , δ, C ) and γ is γ = R n e − ( R − + Z B R − / ∩ Σ {| Hess φ | + | φ |} e − | x | . (1.32)To bound the Hessian term, first choose balls B i = B | zi | ( z i ) so that • B R − / ∩ Σ is contained in the union of the half-balls B i . • Each point is in at most c = c ( n ) of the balls.To simplify notation, set r i = | z i | . Applying Lemma B.1 on B i givessup B i | Hess φ | ≤ C (cid:26) r − n − i Z B i | φ | + (cid:18)Z B i | φ | (cid:19) c ℓ,n (cid:27) , (1.33)where c ℓ,n ∈ (0 ,
1) goes to one as ℓ → ∞ . Note that the Gaussian weight has boundedoscillation on B i (this is why the radius r i was chosen). It follows that Z B R − / ∩ Σ | Hess φ | e − | x | ≤ C X (cid:26) r − i Z B i | φ | + r ni (cid:18)Z B i | φ | (cid:19) c ℓ,n (cid:27) e − | zi | ≤ C R k φ k L ( B R ) + C X (cid:18)Z B i | φ | (cid:19) c ℓ,n e − | zi | (1.34) ≤ C R k φ k L ( B R ) + C k φ k c ℓ,n L ( B R ) , where the last inequality uses the H¨older inequality for sums and the bound for F (Σ). Since k φ k L is bounded (we are interested in the case where it is much less than one), the lowerpower is dominant and we conclude thate −| y | − δ y ≤ C γ ≤ C R n e − ( R − + C R k φ k c ℓ,n L ( B R ) . (1.35) Arguing similarly and using this in the bounds for ∇ τ gives |∇ τ | ( y ) ≤ C R n δ a ℓ,n y ≤ C R n (cid:26) e | y | − ( R − + e | y | k φ k cℓ,n L ( B R ) (cid:27) a ℓ,n , (1.36) (cid:12)(cid:12) ∇ τ (cid:12)(cid:12) ( y ) ≤ C R n +1 δ b ℓ,n y ≤ C R n +22 (cid:26) e | y | − ( R − + e | y | k φ k cℓ,n L ( B R ) (cid:27) b ℓ,n . (1.37) (cid:3) Distance to cylinders and the first Lojasiewicz inequality
In this section, we will prove the first Lojasiewicz inequality that bounds the distancesquared to the space C k of all rotations of the cylinder S k √ k × R n − k by a power close to oneof the gradient of the F functional. This will follow from the bounds on the tensor τ = AH in the previous section together with the following proposition: Proposition 2.1.
Given n , δ > C , there exist ǫ > ǫ > C so that ifΣ ⊂ R n +1 is a hypersurface (possibly with boundary) that satisfies:(1) H ≥ δ > | A | + |∇ A | ≤ C on B R ∩ Σ.(2) B √ n ∩ Σ is ǫ C -close to a cylinder in C k for some k ≥ r ∈ (5 √ n, R ) with r sup B √ n ( | φ | + |∇ φ | ) + r sup B r (cid:0) |∇ τ | + |∇ τ | (cid:1) ≤ ǫ , (2.2)we have that B √ r − k ∩ Σ is the graph over (a subset of) a cylinder in C k of u with(2.3) | u | + |∇ u | ≤ C ( r sup B √ n ( | φ | + |∇ φ | ) + r sup B r (cid:0) |∇ τ | + |∇ τ | (cid:1)) . This proposition shows that Σ must be close to a cylinder as long as H is positive, φ issmall, τ is almost parallel and Σ is close to a cylinder on a fixed small ball. Together withTom Ilmanen, we proved a similar result in proposition 2 . φ ≡
0) and this proposition is inspired by that one.We will prove the proposition over the next two subsections and then turn to the proof ofthe first Lojasiewicz inequality.2.1.
Ingredients in the proof of Proposition 2.1.
This subsection contains the ingre-dients for the proof of Proposition 2.1. The first is the following result from [CIM] (seecorollary 4 .
22 in [CIM]):
Corollary 2.4 ([CIM]) . If Σ ⊂ R n +1 is a hypersurface (possibly with boundary) with • < δ ≤ H on Σ, • the tensor τ ≡ A/H satisfies |∇ τ | + |∇ τ | ≤ ǫ ≤ • At the point p ∈ Σ, τ p has at least two distinct eigenvalues κ = κ ,then | κ κ | ≤ ǫδ (cid:18) | κ − κ | + 1 | κ − κ | (cid:19) . We will use two additional lemmas in the proof of Proposition 2.1. The next lemma showsthat φ controls the distance to the shrinking sphere in a neighborhood of the sphere. This,of course, implies that the shrinking sphere is isolated in the space of shrinkers. The proofuses that the linearized operator is invertible. Lemma 2.5.
Given k and α >
0, there exist ǫ > C so that if Σ ⊂ R k +1 is the graphof a C ,α function u over S k √ k with k u k C ≤ ǫ , then k u k C ,α ≤ C k φ k C α . (2.6) Proof.
On the sphere, the linearized operator L for φ is given by L = ∆ + 1 since | A | = 1 / m -th cluster at m + ( k − m . Scaling this to the sphere of radius √ k ,the m -th cluster is now at m + ( k − m k , (2.7)and, thus, the first three eigenvalues for L = ∆ + 1 occur at − , − and k . In particular, 0is not an eigenvalue and, thus, L is invertible and, by the Schauder estimates, we have k u k C ,α ≤ C k L u k C α , (2.8)where C depends only on k and α . The lemma follows from this and the fact that thelinearization of φ is L and the error is quadratic (cf. Lemma 4.10 below) so we have k φ − Lu k C α ≤ C k u k C k u k C ,α , (2.9)where C again depends only on k and α . Combining the last two inequalities gives k u k C ,α ≤ C k φ k C α + C k u k C k u k C ,α ≤ C k φ k C α + C ǫ k u k C ,α , (2.10)which gives the claim after choosing ǫ > C ǫ = . (cid:3) The next lemma shows if Σ has an approximate translation and is almost a shrinker, thenslicing Σ orthogonally to the translation gives a submanifold Σ of one dimension less thatis also almost a shrinker. We will use this to repeatedly slice an almost cylinder to get downto the almost sphere. We let φ be the φ of Σ (so Σ ⊂ R k is a shrinker when φ ≡ Lemma 2.11.
Let Σ ⊂ R k +1 be a hypersurface, Σ = { x k +1 = 0 } ∩ Σ, and x ∈ Σ a pointwhere Σ intersects the hyperplane { x k +1 = 0 } transversely. If we have • (cid:12)(cid:12) ∇ T x k +1 (cid:12)(cid:12) ≥ − ǫ > / • (cid:12)(cid:12) ∇ T ∇ T x k +1 (cid:12)(cid:12) ≤ ǫ , • (cid:12)(cid:12) A ( · , ∇ T x k +1 ) (cid:12)(cid:12) + (cid:12)(cid:12) ( ∇ A ) ( · , ∇ T x k +1 ) (cid:12)(cid:12) ≤ ǫ .Then at x | φ − φ | + |∇ Σ ( φ − φ ) | ≤ ǫ { | φ | + |∇ φ |} . (2.12) Proof.
Set v = ∇ T x k +1 = ∂ Tk +1 . Let e , . . . , e k − be an orthonormal frame for Σ , so that e , . . . , e k − , v | v | (2.13) gives an orthonormal frame for Σ. If n ∈ R k +1 and n ∈ R k denote the normals to Σ andΣ , respectively, then n = | v | n + h ∂ k +1 , n i ∂ k +1 . (2.14)(To see this, check that this unit vector is orthogonal to the frame (2.13).) Since h∇ e i e j , ∂ k +1 i =0, the expression for n gives h∇ e i e j , n i = | v | h∇ e i e j , n i . It follows that H − H = − (cid:26) A ( e i , e i ) + A (cid:18) v | v | , v | v | (cid:19)(cid:27) + h∇ e i e i , n i = 1 − | v || v | A ( e i , e i ) − A (cid:18) v | v | , v | v | (cid:19) = | v | − | v | H − | v | A (cid:18) v | v | , v | v | (cid:19) . (2.15)Similarly, given x ∈ Σ , we have x k +1 = 0 and, thus, h x, n i − h x , n i = h x, n i − h x, n i = | v | − | v | h x, n i . (2.16)Combining the last two equations gives for x ∈ Σ that φ − φ = 12 ( h x, n i − h x , n i ) − ( H − H ) = | v | − | v | (cid:26) h x, n i − H (cid:27) + 1 | v | A (cid:18) v | v | , v | v | (cid:19) = | v | − | v | φ + 1 | v | A (cid:18) v | v | , v | v | (cid:19) . (2.17)Since | v | ≥ / − | v | ≤ ǫ , it follows that | φ − φ | ≤ ǫ | φ | + 8 | A ( v, v ) | ≤ ǫ | φ | + 8 ǫ . (2.18)Similarly, we bound the derivative by |∇ ( φ − φ ) | ≤ − | v | ) |∇ φ | + 2 |∇ v | | φ | + 4 (1 − | v | ) |∇ v | | φ | + 16 |∇ v | | A ( v, v ) | + 8 |∇ A ( v, v ) | + 16 | A ( v, ∇ v ) | (2.19) ≤ ǫ |∇ φ | + 4 ǫ | φ | + 16 ǫ . (cid:3) The proof of Proposition 2.1.
Proof of Proposition 2.1.
Within the proof, it will be convenient to set ǫ τ ( r ) = sup B r (cid:0) |∇ τ | + |∇ τ | (cid:1) and ǫ φ ( r ) = sup B r ( | φ | + |∇ φ | ) . (2.20) Step 1: The approximate translations.
Using the C -closeness in (2), at every p inΣ ∩ B √ n there are n − k orthonormal eigenvectors v ( p ) , . . . , v n − k ( p ) , of A with eigenvalues κ , . . . κ n − k with absolute value less than 1 / √ n , plus k ≥ σ , . . . σ k with absolute value at least 1 / √ n . By (1), we can applyCorollary 2.4 to obtain | κ j ( p ) | ≤ C ǫ τ (5 √ n ) , j = 1 , . . . , n − k, (2.21)where C depends only on n and δ . Now fix some p in Σ ∩ B √ n and define n − k linear functions f i on R n +1 and tangentialvector fields v i on Σ by f i ( x ) = h v i ( p ) , x i and v i = ∇ T f i = v i ( p ) − h v i ( p ) , n i n . Step 2: Extending the bounds away from p . For each r > √ n , let Ω r denote theset of points in B r ∩ Σ that can be reached from p by a path in B r ∩ Σ of length at most 3 r .The v i ’s have the following three properties on Ω r : | v i − v i ( p ) | ≤ C r ǫ τ ( r ) , (2.22) | τ ( v i ) | ≤ C r ǫ τ ( r ) , (2.23) |∇ v i A | ≤ C r ǫ τ ( r ) , (2.24)where C depends only on n , δ and C .To prove (2.22) and (2.23), suppose that γ : [0 , r ] → Σ is a curve with γ (0) = p and | γ ′ | ≤ w is a parallel unit vector field along γ with w (0) = v i ( p ). Therefore, thebound on ∇ τ gives |∇ γ ′ τ ( w ) | ≤ ǫ τ ( r ) and, thus, | τ ( w ) | ≤ r ǫ τ ( r ) + | τ p ( v i ( p )) | ≤ ( C + 3 r ) ǫ τ ( r ) ≤ C r ǫ τ ( r ) . (2.25)In particular, we also have | A ( w ) | = | H | | τ ( w ) | ≤ C r ǫ τ ( r ) . (2.26)Therefore, since ∇ R n +1 γ ′ w = A ( γ ′ , w ) n , the fundamental theorem of calculus gives | w ( t ) − v i ( p ) | = | w ( t ) − w (0) | ≤ Z r | A ( w ( s )) | ds ≤ C r ǫ τ ( r ) . (2.27)Since w ( t ) is tangential, we see that |h v i ( p ) , n i| ≤ C r ǫ τ ( r ), giving (2.22). Similarly, (2.27)gives that | w ( t ) − v i | = (cid:12)(cid:12)(cid:12) ( w ( t ) − v i ( p )) T (cid:12)(cid:12)(cid:12) ≤ | w ( t ) − v i ( p ) | ≤ C r ǫ τ ( r ) . (2.28)If we combine this (and the boundedness of τ ) with (2.25), the triangle inequality gives | τ ( v i ) | ≤ | τ ( w ) | + | τ ( w − v i ) | ≤ C r ǫ τ ( r ) , (2.29)where we used the lower bound on r to bound r by r . This gives (2.23).We will see that (2.23) implies (2.24). Namely, given unit vector fields x and y , the Codazziequation gives | ( ∇ v i A ) ( x, y ) | = | ( ∇ x A ) ( v i , y ) | = | ( ∇ x ( H τ )) ( v i , y ) | = | H ( ∇ x τ ) ( v i , y ) | + | ( ∇ x H ) τ ( v i , y ) | ≤ C ǫ τ ( r ) + C r ǫ τ , (2.30)where the last inequality used that | H | and |∇ H | are bounded by (1). This gives (2.24). Step 3: The sphere.
From the ǫ closeness to C k in B √ n in (2), we know thatΣ ≡ B √ n ∩ Σ ∩ { f = · · · = f n − k = 0 } is a compact topological S k of radius fixed close to √ k . Using (2.22)–(2.24), we can applyLemma 2.11 ( n − k ) times to get that Σ has k φ k C ≤ C ( ǫ τ + ǫ φ ) , (2.31) where ǫ τ and ǫ φ are evaluated at r = 5 √ n . We can now apply Lemma 2.5 to get that Σ isa graph over S k √ k of a function u with k u k C ,α ≤ C ( ǫ τ + ǫ φ ) . (2.32) Step 4: The translations and extending the bound.
Let y , . . . , y k +1 be an or-thonormal basis of linear functions orthogonal to the f i ’s. Define the function w by w ≡ k +1 X i =1 y i , (2.33)so that w would be identically equal to √ k if Σ was in C k . In our case, it follows from (2.32)that the restriction w of w to Σ satisfies k w − √ k k C ,α (Σ ) ≤ C ( ǫ τ + ǫ φ ) . (2.34)We will use the v j ’s to extend the bounds away from Σ inside Ω r . Namely, for each y i and v j and any point in Ω r , we have (cid:12)(cid:12) ∇ v j ∇ T y i (cid:12)(cid:12) = (cid:12)(cid:12) ∇ v j ∇ ⊥ y i (cid:12)(cid:12) ≤ | A ( v j , · ) | ≤ C r ǫ τ ( r ) , (2.35)where the last inequality used (2.23) and the positive lower bound for H .We will extend the bounds by constructing a “radial flow”. First, define a function f by f = n − k X i =1 f i , and then define the vector field v by v = ∇ T f |∇ T f | . Thus, the flow by v preserves the level sets of f . Note that ∇ T f = P f i ∇ T f i f = X f i f v i = X f i f v i ( p ) + X f i f ( v i − v i ( p )) . (2.36)Since the v i ( p )’s are orthonormal and P (cid:16) f i f (cid:17) = 1, it follows that (cid:12)(cid:12)(cid:12)(cid:12)X f i f v i ( p ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . Combining this with the triangle inequality and (2.22) gives thatsup Ω r (cid:12)(cid:12) − (cid:12)(cid:12) ∇ T f (cid:12)(cid:12)(cid:12)(cid:12) ≤ X | v i ( p ) − v i | ≤ C r ǫ τ ( r ) , (2.37)where C depends only on n , δ and C . We will assume from now on that r satisfies C r ǫ τ ( r ) ≤ , (2.38) so that (cid:12)(cid:12) − (cid:12)(cid:12) ∇ T f (cid:12)(cid:12)(cid:12)(cid:12) ≤ and, thus, that sup Ω r | v | ≤
2. Since v is in the span of the v i ’s and | v | ≤
2, it follows from (2.35) thatsup Ω r (cid:12)(cid:12) ∇ v ∇ T y i (cid:12)(cid:12) ≤ C r ǫ τ ( r ) . (2.39)Since h∇ y i , v j i = 0 at p and | v i − v i ( p ) | ≤ C r ǫ τ ( r ) on Ω r by (2.22), we know that (cid:12)(cid:12)(cid:12) ∇ Tv j y i (cid:12)(cid:12)(cid:12) ≤ C r ǫ τ ( r ) on Ω r . Hence, since v is in the span of the v j ’s and | v | ≤ |∇ v y i | ≤ C r ǫ τ ( r ) on Ω r . Combiningthis and (2.39) givessup Ω r (cid:12)(cid:12) ∇ v ∇ T w (cid:12)(cid:12) = 2 sup Ω r (cid:12)(cid:12) ∇ v ( y i ∇ T y i ) (cid:12)(cid:12) ≤ k + 1) sup Ω r (cid:8) |∇ v y i | |∇ T y i | + | y i | (cid:12)(cid:12) ∇ v ∇ T y i (cid:12)(cid:12)(cid:9) ≤ C r ǫ τ ( r ) . (2.40)We will now define a subset Ω r,f of Ω r given by flowing Σ outwards along the vector field v . To do this, let Φ( q, t ) to be the flow by v at time t starting from q and setΩ r,f = (cid:8) Φ( q, t ) | q ∈ Σ , t ≤ r − k and Φ( q, s ) ∈ Ω r for all s ≤ t (cid:9) . (2.41)By integrating (2.40) up from Σ , we conclude thatsup Ω r,f (cid:12)(cid:12) ∇ T w (cid:12)(cid:12) ≤ sup Σ (cid:12)(cid:12) ∇ T w (cid:12)(cid:12) + 6 r sup Ω r (cid:12)(cid:12) ∇ v ∇ T w (cid:12)(cid:12) ≤ C ǫ φ + C r ǫ τ ( r ) . (2.42)Integrating (2.42) from Σ gives thatsup Ω r,f (cid:12)(cid:12) w − k (cid:12)(cid:12) ≤ C r ǫ φ + C r ǫ τ ( r ) . (2.43)Observe next that as long as C r ǫ φ + C r ǫ τ ( r ) ≤ k , (2.44)then we can conclude that Ω r,f = { f ≤ r − k } ∩ Σ . (2.45)This gives a positive lower bound for w on Ω r,f so the bound on ∇ T w then givessup Ω r,f (cid:12)(cid:12) ∇ T w (cid:12)(cid:12) ≤ C ǫ φ + C r ǫ τ ( r ) , (2.46)so the C bound on w , and thus also on u , hold as claimed. (cid:3) Proving the first Lojasiewicz inequality.
In this subsection, we will prove Theorem0.24. The proof not only gives the L closeness to a cylinder, but also gives pointwise closenesson a scale that depends on φ and the initial graphical scale of Σ. Proof of Theorem 0.24.
We have that B R ∩ Σ is a smooth graph over a cylinder of a function¯ u with k ¯ u k C ,α ≤ ǫ and |∇ ℓ ¯ u | ≤ C ℓ and that Σ satisfies:(1) H ≥ δ > | A | + |∇ A | ≤ C on B R ∩ Σ.(2) B √ n ∩ Σ is ǫ C -close to a cylinder in C k for some k ≥ The starting point is Proposition 2.1 which gives, for any r ∈ (5 √ n, R ) with r sup B √ n ( | φ | + |∇ φ | ) + r sup B r (cid:0) |∇ τ | + |∇ τ | (cid:1) ≤ ǫ , (2.47)we have that B √ r − k ∩ Σ is the graph over (a subset of) a cylinder in C k of u with(2.48) | u | + |∇ u | ≤ C ( r sup B √ n ( | φ | + |∇ φ | ) + r sup B r (cid:0) |∇ τ | + |∇ τ | (cid:1)) . Using the a priori bounds and assuming that ℓ is large enough, we can use the interpolationinequalities of Lemma B.1 to get thatsup B √ n ( | φ | + |∇ φ | ) ≤ C k φ k L ( B R ) , (2.49)where C = C ( n ) and L ( B R ) denotes the Gaussian L norm on B R .To get bounds on ∇ τ and ∇ τ , we apply Corollary 1.27 to get C = C ( n, λ , ℓ, C ℓ ) sothat for r + r < R −
1, we havesup B r (cid:0) |∇ τ | + |∇ τ | (cid:1) ≤ C R n (cid:26) e − d ℓ,n ( R − + k φ k dℓ,n L ( B R ) (cid:27) e r , (2.50)where the exponent d ℓ,n ∈ (0 ,
1) has lim ℓ →∞ d ℓ,n = 1.Thus, we see that B √ r − k ∩ Σ is the graph over (a subset of) a cylinder Σ k ∈ C k of u with | u | + |∇ u | ≤ C (cid:26) r k φ k L + r R n (cid:26) e − d ℓ,n ( R − + k φ k dℓ,n L ( B R ) (cid:27) e r (cid:27) ≤ C R n +5 (cid:26) e − d ℓ,n ( R − + k φ k dℓ,n L ( B R ) (cid:27) e r , (2.51)where C = C ( n, λ , ℓ, C ℓ ) and this holds so long as the right hand side is at most ǫ > R ≤ R − L bound, we first use (2.51) on B R to get Z B R (cid:12)(cid:12)(cid:12) w Σ k − √ k (cid:12)(cid:12)(cid:12) e − | x | ≤ C R n +10 (cid:26) e − d ℓ,n ( R − + k φ k d ℓ,n L ( B R ) (cid:27) , (2.52)and then use that (cid:12)(cid:12)(cid:12) w Σ k − √ k (cid:12)(cid:12)(cid:12) ( x ) ≤ | x | to get that Z B R \ B R (cid:12)(cid:12)(cid:12) w Σ k − √ k (cid:12)(cid:12)(cid:12) e − | x | ≤ C R n +2 e − R ≤ C R n +12 (cid:26) e − d ℓ,n ( R − + k φ k d ℓ,n L ( B R ) (cid:27) , (2.53)where the last inequality is the definition of R . Combining these completes the proof. (cid:3) We will later also need a variation on this, where we assume bounds on A and H on alarge scale and conclude that Σ is a graph over a cylinder on a large set. Theorem 2.54.
There exist R , ℓ and δ > ⊂ R n +1 has λ (Σ) ≤ λ and We will take ℓ large later; we could replace 3 / ℓ larger. (1) for some R > R , we have on B R ∩ Σ that | A | + |∇ ℓ A | ≤ C and H ≥ δ > B R ∩ Σ is a C graph over some cylinder in C k with norm at most δ .Then there is a cylinder ˜Σ ∈ C k so that(3) B R − ∩ Σ is the graph of u over ˜Σ with k u k C ,α ≤ ǫ ,where R is given by R = max (cid:26) r ≤ R − (cid:12)(cid:12) R n +5 (cid:18) e − b ℓ ,n ( R − + k φ k bℓ ,n L ( B R ) (cid:19) e r ≤ ˜ C (cid:27) , (2.55)the exponent b ℓ ,n ∈ (0 ,
1) satisfies lim ℓ →∞ b ℓ ,n = 1 and ˜ C = ˜ C ( n, λ , δ , C ). Proof.
We follow the proof of Theorem 0.24 up through (2.51) to get ˜Σ ∈ C k and a function u so that B R − ∩ Σ is the graph of u over ˜Σ, R is defined by (2.55), and | u | + |∇ u | ≤ δ . (2.56)Finally, we use interpolation and the ∇ ℓ A bound to get the desired C ,α bound when δ > (cid:3) Analysis on the cylinder
In this section, we will prove estimates for the L and L operators on a cylinder Σ ∈ C k with k ∈ { , . . . , n − } . These estimates will be used in the next section to prove our secondLojasiewicz inequality. Note that L = L + 1 on Σ since | A | ≡ .We will use the Gaussian L -norm k u k L = R u e − | x | , as well as the associated Gaussian W , and W , norms k u k W , = Z (cid:0) u + |∇ u | (cid:1) e − | x | and k u k W , = Z (cid:0) u + |∇ u | + | Hess u | (cid:1) e − | x | . (3.1)3.1. Symmetry, the spectrum of L and a Poincar´e inequality. The starting point isthe following elementary lemma that summarizes the key properties of the L operator onΣ ∈ C k : Lemma 3.2.
The operator L on Σ is symmetric on W , with Z Σ u L v e − | x | = − Z Σ h∇ u, ∇ v i e − | x | . (3.3)The space W , embeds compactly into L and L has discrete spectrum with finite multi-plicity on W , with a complete basis of smooth L -orthonormal eigenfunctions. Proof.
The first claim follows from integration by parts. The second follows from [BE] sinceΣ has positive Bakry- ´Emery Ricci curvature and finite weighted volume. Finally, the lastclaim is a consequence of the first two (cf. theorem 10 .
20 in [Gr]). (cid:3)
We will also use the following Gaussian Poincar´e inequality on Σ = S k √ k × R n − k . Themiddle term does not use the full gradient, but only the gradient in the translation directions. Lemma 3.4.
There exists C = C ( k, n ) so that if Σ ∈ C k and u ∈ W , , then k| x | u k L ≤ C (cid:0) k u k L + k∇ R n − k u k L (cid:1) ≤ C k u k W , . (3.5) Proof.
Let y be coordinates on the R n − k factor, so that x T = y and | x | = | y | + 2 k . (3.6)We compute e | x | div Σ (cid:18) u y e − | x | (cid:19) = 2 u h∇ u, y i + ( n − k ) u − u | y | ≤ |∇ R n − k u | + ( n − k ) u − u | y | , (3.7)where the inequality used the absorbing inequality 2 ab ≤ a + 4 b .By approximation, we can assume that u has compact support on Σ and, thus, Stokes’theorem gives 14 Z Σ u | y | e − | x | ≤ Z Σ (cid:8) ( n − k ) u + 4 |∇ R n − k u | (cid:9) e − | x | . (3.8)The lemma follows since u | x | = u ( | y | + 2 k ). (cid:3) Estimates for the projection onto the kernel of L . Let K be the kernel of L K = { v ∈ W , | Lv = 0 } . (3.9)Given any u ∈ W , , we let u K denote the L -orthogonal projection of u onto K and u ⊥ = u − u K (3.10)the projection onto the L -orthogonal complement of K .The next lemma shows that L is bounded from W , to L , L is uniformly invertible on K ⊥ and the projection onto K is bounded from L to W , . Lemma 3.11.
Given n , there exist C and µ > C k k Lu k L ≤ C k u k W , , (3.12) µ k u ⊥ k W , ≤ k Lu k L , (3.13) k u K k W , ≤ C k u k L . (3.14) Proof.
Since L = ∆ + ∇ x T + 1 on the cylinder, we have k Lu k L ≤ k ∆ u k L + k u k L + 12 k| x | |∇ u |k L . (3.15)The first claim follows from this and using Lemma 3.4 to get the bound k| x | |∇ u |k L ≤ C k|∇ u |k W , ≤ C {k|∇ u |k L + k Hess u k L } . (3.16)To get the second claim, we will need the “Gaussian elliptic estimate” k v k W , ≤ C ( k v k L + kL v k L ) , (3.17)where C depends on n and the estimate holds for any v ∈ W , . To prove (3.17), we firstintegrate by parts to get k∇ v k L = |h v, L v i L | ≤ k v k L kL v k L ≤ k v k L + 12 kL v k L . (3.18) Thus, we see that k v k W , is bounded by the right hand side of (3.17). It remains to boundthe L norm of the Hessian of v . This will follow from what we’ve done and the divergencetheorem sincee | x | div Σ (cid:18) { v ij v i − ( L v ) v j } e − | x | (cid:19) = 12 L |∇ v | − ( L v ) − h∇L v, ∇ v i≥ | Hess v | − ( L v ) , (3.19)where the last inequality used the Bochner formula for the drift Laplacian on the cylinder. The second claim now follows by first applying Lemma 3.2 to get µ > µ k u ⊥ k L ≤ k Lu ⊥ k L = k Lu k L (3.20)and then using (3.17) to bound the W , norm.The final claim follows from the trivial projection bound k u K k L ≤ k u k L and the bound k u K k W , ≤ C k u K k L . (3.21)To see (3.21), first use the equation L u K = − u K to get k∇ u K k L = k u K k L , and then usethe Bochner formula as in (3.19) to bound the Hessian of u K in terms of k u K k W , . (cid:3) We will also need the next lemma that bounds the Gaussian L norm of a quadraticexpression in u, ∇ u, Hess u that bounds the error term in the linear approximation of thegradient of the F functional. When u ∈ K , the bound is the square of the Gaussian L norm while we obtain a weaker bound when u is orthogonal to K . Lemma 3.22.
There exist C K = C K ( n ) and C = C ( n ) so that if u ∈ W , , then (cid:13)(cid:13)(cid:13) u K + |∇ u K | + (cid:12)(cid:12) Hess u K ( · , R n − k ) (cid:12)(cid:12) + (1 + | x | ) − | Hess u K | (cid:13)(cid:13)(cid:13) L ≤ C K k u K k L , (3.23) (cid:13)(cid:13)(cid:13) ( u ⊥ ) + |∇ u ⊥ | + (cid:12)(cid:12) Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) + (1 + | x | ) − | Hess u ⊥ | (cid:13)(cid:13)(cid:13) L ≤ C k u k C k u ⊥ k W , . (3.24)The key for proving both claims is an explicit description of K . Namely, K is generated bymultiplying a polynomial eigenfunction of L R n − k times a spherical eigenfunction of ∆ S k √ k .To state this, let y i be coordinates on the R n − k factor and let θ be in the S k factor. Lemma 3.25.
Each v ∈ K can be written as v ( y, θ ) = q ( y ) + X i y i f i ( θ ) + c , (3.26)where q is a homogeneous quadratic polynomial on R n − k , each f i is an eigenfunction on S k √ k with eigenvalue , and c is a constant. Proof.
The operator L splits as L = L + 1 = ∆ θ + L y + 1 , (3.27)where ∆ θ is the Laplacian on S k √ k and L y is the drift operator on R n − k . See, for instance, (6.7) below. This would be obvious if the C norm of v were bounded by the L norm, but this is not the case. The first obervation is that differentiating with respect to y i lowers the eigenvalue by .Thus, if we set v i = ∂v∂y i and v ij = ∂ v∂y i ∂y j , then L v i = − v i , (3.28) L v ij = 0 . (3.29)Since every L L -harmonic function must be constant, we conclude that v ij is constant. Asa consequence, the function v can be written as v = X i,j a ij y i y j + X i f i ( θ ) y i + g ( θ ) , (3.30)where a ij ∈ R , each f i is a function on S k √ k and g is a function on S k √ k .Note that L y y i = − y i , (3.31) L y ( y i y j ) = 2 δ ij − y i y j . (3.32)Using this and the decomposition of L from (3.27), we get that0 = Lv = X i,j a ij (2 δ ij ) + X i (cid:20) y i ∆ θ f i ( θ ) + 12 f i ( θ ) y i (cid:21) + (∆ θ + 1) g ( θ ) . (3.33)Observe first that only the middle terms depend on y . Setting these equal to zero, weconclude that each f i satisfies ∆ θ f i = − f i . (3.34)It follows that g + 2 P a ii is a S k √ k eigenfunction with eigenvalue one, i.e.,∆ θ (cid:16) g + 2 X a ii (cid:17) = − (cid:16) g + 2 X a ii (cid:17) . (3.35)However, one is not an eigenvalue of ∆ θ (the eigenvalues jump from 1 / k + 1) /k ; see(2.7)), so we have g ≡ − P a ii . (cid:3) It is interesting to note that the y i f i part of K corresponds to rotations. However, by[CIM], the quadratic polynomials in the kernel are not generated by one-parameter familiesof shrinkers. In particular, the kernel K contains non-integrable functions.As a corollary of Lemma 3.25, we get C pointwise estimates for functions in the kernelof L that grow at most quadratically in | y | : Corollary 3.36.
There exists C depending on n so that if v ∈ K , thensup | v | ≤ C (1 + | y | ) k v k L , (3.37) sup |∇ v | ≤ C (1 + | y | ) k v k L , (3.38) sup (cid:12)(cid:12) Hess v ( · , R n − k ) (cid:12)(cid:12) ≤ C k v k L , (3.39) sup | Hess v | ≤ C (1 + | y | ) k v k L . (3.40) Remark 3.41.
The point of (3.39) is that, as opposed to (3.40), we get a better bound,that does not grow in y , if we restrict to the Hessian in the Euclidean factor. This is usefullater. Proof of Corollary 3.36.
Since K is finite dimensional, the estimates (3.37)–(3.40) will followfor all of K from the squared triangle inequality once we show that there is an orthogonalbasis for K where each element in the basis satisfies (3.37)–(3.40).The key for this is Lemma 3.25 which shows that K can be written as K = K ⊕ K , where(3.42) • Each v ∈ K is given by P i y i f i where f i is a S k √ k eigenfunction with eigenvalue . • Each v ∈ K is a constant plus a homogeneous quadratic polynomial in y .In particular, (3.42) is a L -orthogonal decomposition. Case 1 . If f i , f j are S k √ k eigenfunctions with eigenvalue , then h y i f i , y j f j i L = 0 if i = j , (3.43)so we get an orthogonal basis for K consisting of a single y i times an f . Suppose that v = y i f , (3.44)where f is a S k √ k eigenfunction with eigenvalue . Note that k v k L = e − k k f k L θ Z R n − k y i e − | y | dy ≡ C k k f k L θ , (3.45)where the constant C k > k and the sub θ denotes the norms on S k √ k .Using elliptic estimates for the compact manifold S k √ k , we have c = c ( k ) so that k f k C θ ≤ c k f k L θ . (3.46)Therefore, at each point, we have that | v | = y i f ≤ c y i k f k L θ , (3.47) |∇ v | = y i |∇ θ f | + f ≤ c (cid:0) y i (cid:1) k f k L θ , (3.48) | Hess v | = y i | Hess f | ≤ c y i k f k L θ , (3.49) (cid:12)(cid:12) Hess v ( · , R n − k ) (cid:12)(cid:12) ≤ |∇ θ f | ≤ c , (3.50)giving the desired bounds in this case (the first bound is even better than needed). Case 2 . It is easy to see that an orthogonal basis for K is given by { y i y j − δ ij | i ≤ j } . (3.51)Therefore, it suffices to show (3.37)–(3.40) when v = y i y j − δ ij . (3.52)However, this follows immediately since the L norms are nonzero and v is a quadraticpolynomial in y (in this case, the Hessian bound is even better than needed). (cid:3) We will now use the estimates from the corollary to prove Lemma 3.22. Proof of Lemma 3.22.
To simplify notation, set k v k ≡ (cid:13)(cid:13)(cid:13) v + |∇ v | + (cid:12)(cid:12) Hess v ( · , R n − k ) (cid:12)(cid:12) + (1 + | x | ) − | Hess v | (cid:13)(cid:13)(cid:13) L . (3.53)Given a ∈ R , note that k a v k = a k v k .We will show that there is a constant C K so that C K ≡ sup (cid:8) k w k (cid:12)(cid:12) w ∈ K and k w k L = 1 (cid:9) < ∞ . (3.54)Once we have this, then for a general v ∈ K , we set w = v k v k L so that k v k = k k v k L w k = k v k L k w k ≤ C K k v k L , (3.55)giving the first claim (3.23).To establish (3.54), apply Corollary 3.36 to get C = C ( n ) so that | w | + |∇ w | + | Hess w | ≤ C (1 + | y | ) . (3.56)Integrating this polynomially growing bound against the exponential decaying Gaussianweight gives the desired uniform bound on k w k .To prove (3.24), we will show that (cid:13)(cid:13) ( u ⊥ ) (cid:13)(cid:13) L , (cid:13)(cid:13) |∇ u ⊥ | (cid:13)(cid:13) L , (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) L and (cid:13)(cid:13) (1 + | x | ) − | Hess u ⊥ | (cid:13)(cid:13) L (3.57)are each bounded by C k u k C k u ⊥ k W , for a constant C depending only on the dimension n . The key point will be the bounds (3.37)–(3.40) on u K from Corollary 3.36.For the first term, we use (3.37) to get( u ⊥ ) = ( u − u K ) u ⊥ ≤ (cid:0) k u k C + C (1 + | x | ) k u K k L (cid:1) | u ⊥ |≤ C k u k C (1 + | x | ) | u ⊥ | , (3.58)where the last inequality used the projection inequality k u K k L ≤ k u k L and the trivialinequality k u k L ≤ C k u k C that follows since Σ has finite Gaussian area. Integrating andapplying Lemma 3.4 twice gives k ( u ⊥ ) k L ≤ C k u k C k (1 + | x | ) u ⊥ k L ≤ C k u k C k u ⊥ k W , . (3.59)For the second term, we use the triangle inequality and (3.38) to get |∇ u ⊥ | ≤ ( |∇ u | + |∇ u K | ) |∇ u ⊥ | ≤ ( k u k C + C (1 + | x | ) k u K k L ) |∇ u ⊥ |≤ C k u k C (1 + | x | ) |∇ u ⊥ | , (3.60)where the last inequality follows as above. Integrating and applying Lemma 3.4 gives k|∇ u ⊥ | k L ≤ C k u k C k (1 + | x | ) |∇ u ⊥ |k L ≤ C k u k C k u ⊥ k W , . (3.61)For the third term, we use the triangle inequality and (3.39) to get (cid:12)(cid:12) Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) ≤ (cid:8)(cid:12)(cid:12) Hess u ( · , R n − k ) (cid:12)(cid:12) + (cid:12)(cid:12) Hess u K ( · , R n − k ) (cid:12)(cid:12)(cid:9) (cid:12)(cid:12) Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) ≤ {k u k C + C k u K k L } (cid:12)(cid:12) Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) (3.62) ≤ C k u k C (cid:12)(cid:12) Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) . Integrating this gives (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)
Hess u ⊥ ( · , R n − k ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) L ≤ C k u k C k Hess u ⊥ k L . (3.63)Finally, for the fourth (last) term, we use the triangle inequality and (3.40) to get(1 + | x | ) − | Hess u ⊥ | ≤ (1 + | x | ) − ( | Hess u | + | Hess u K | ) | Hess u ⊥ |≤ (1 + | x | ) − {k u k C + C (1 + | x | ) k u K k L } | Hess u ⊥ | (3.64) ≤ C k u k C | Hess u ⊥ | . To bound the last term and complete the proof of (3.24), we integrate this to get (cid:13)(cid:13) (1 + | x | ) − | Hess u ⊥ | (cid:13)(cid:13) L ≤ C k u k C k Hess u ⊥ k L . (cid:3) The gradient Lojasiewicz inequality for F In this section, we will prove a gradient Lojasiewicz inequality for F in a neighborhood ofa cylinder Σ ∈ C k . The inequality will hold for graphs over part of Σ with small C norm.The key technical ingredient is the next proposition which shows that our first Lojasiewiczimplies our gradient Lojasiewicz inequality. Proposition 4.1.
There exist C = C ( n, λ ) and ¯ ǫ = ¯ ǫ ( n ) > λ (Σ) ≤ λ and B ˜ R ∩ Σ is the graph of ˜ u over a cylinder in C k with k ˜ u k C ≤ ¯ ǫ , then for any β ∈ [0 , | F (Σ) − F ( C k ) | ≤ C k φ k β L ( B ˜ R ) + C (1 + ˜ R n − ) e − (3+ β )( ˜ R − + C k ˜ u k β β L ( B ˜ R ) . (4.2)The proof of Proposition 4.1 is an infinite dimensional version of the model argumentusing Taylor expansion given in Subsection 0.4. The simple model was done with β = 0, butwould have worked with any β ∈ [0 , β close to one. The linearization of the gradient of the F functional. Given a graph Σ u of afunction u over a cylinder Σ ∈ C k , we let F ( u ) ≡ F (Σ u ) and then let M ( u ) be the gradientof F . The next lemma gives linear and quadratic approximations for M and F , respectively. Lemma 4.3.
There exists C so that if the C norm of u is sufficiently small and u is definedon the entire cylinder, then kM ( u ) − Lu k L ≤ C (cid:13)(cid:13) u + |∇ u | + |∇ R n − k |∇ u || + (1 + | x | ) − | Hess u | (cid:13)(cid:13) L , (4.4) (cid:12)(cid:12)(cid:12)(cid:12) F ( u ) − F ( C k ) − h u, Lu i L (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k u k L (cid:13)(cid:13) u + |∇ u | + |∇ R n − k |∇ u || + (1 + | x | ) − | Hess u | (cid:13)(cid:13) L . (4.5) The k φ k β L ( B ˜ R ) term is fine for any β > k ˜ u k β β L ( B ˜ R ) term is fine if β < The bound in the first inequality in Lemma 4.3 is essentially quadratic in u . For example,it is bounded by C k u k C k u k W , . Ideally, we would have liked the bound to be quadratic in k u k W , , but the exponential decay in the Gaussian norm makes this impossible and, thus,leads to technical complications.We will prove Lemma 4.3 in this subsection. The starting point is the next lemma com-puting M ( u ) in terms of u, ∇ u and Hess u . Lemma 4.6.
If Σ is a cylinder in C k and p ∈ Σ, then M ( u )( p ) = f ( u ( p ) , ∇ u ( p )) + h p, V ( u ( p ) , ∇ u ( p )) i + Φ αβ ( u ( p ) , ∇ u ( p )) u αβ ( p ) , (4.7)where f , V and Φ αβ depend smoothly on ( s, y ) for | s | small.Lemma 4.6 is proven in Appendix A.The next lemma shows, for a general map u → M ( u ) of the form (4.7), that the lineariza-tion gives a good approximation up to quadratic error. To state this precisely, consider ageneral map N ( u ) of the form N ( u )( p ) = f ( p, u ( p ) , ∇ u ( p )) + Φ αβ ( p, u ( p ) , ∇ u ( p )) u αβ ( p ) , (4.8)where f and Φ αβ are smooth functions of ( p, s, y ) where p is the point, s ∈ R , and y is atangent vector at p . The linearization of N at u is defined to be L u v = ddt (cid:12)(cid:12) t =0 N ( u + tv ) = f s v + f y α v α + Φ αβ v αβ + u αβ (cid:16) Φ αβs v + Φ αβy γ v γ (cid:17) , (4.9)where all functions are evaluated at the same point p and we have left out the obviousdependence of f and Φ on ( p, u ( p ) , ∇ u ( p )). Lemma 4.10. If N ( u ) is given by (4.8), then we get at each point p that |N ( u + v ) − N ( u ) − L u v | ≤ C ( | v | + |∇ v | ) + C ( | v | + |∇ v | ) | Hess v | , (4.11)where the constants C = C ( p ) and C = C ( p ) are given by C = Lip p ( f s ) + | u αβ | Lip p (cid:0) Φ αβs (cid:1) + Lip p ( f y γ ) + | u αβ | Lip p (cid:16) Φ αβy γ (cid:17) , (4.12) C = (cid:12)(cid:12) Φ αβs (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Φ αβy γ (cid:12)(cid:12)(cid:12) + Lip p (cid:0) Φ αβ (cid:1) . (4.13)Here Lip p denotes the Lipschitz norm at p with respect to the s and y variables. Proof.
Using (4.9), we get at p that for any w that | L u + w v − L u v | ≤ | f s ( p, u + w, ∇ u + ∇ w ) − f s ( p, u, ∇ u ) | | v | + (cid:12)(cid:12) ( u αβ + w αβ ) Φ αβs ( p, u + w, ∇ u + ∇ w ) − u αβ Φ αβs ( p, u, ∇ u ) (cid:12)(cid:12) | v | + | f y α ( p, u + w, ∇ u + ∇ w ) − f y α ( p, u, ∇ u ) | | v α | + (cid:12)(cid:12)(cid:12) ( u αβ + w αβ ) Φ αβy γ ( p, u + w, ∇ u + ∇ w ) − u αβ Φ αβy γ ( p, u, ∇ u ) (cid:12)(cid:12)(cid:12) | v γ | + (cid:12)(cid:12) Φ αβ ( p, u + w, ∇ u + ∇ w ) − Φ αβ ( p, u, ∇ u ) (cid:12)(cid:12) | v αβ | . (4.14) Bounding these terms gives | L u + w v − L u v | ≤ (cid:8)(cid:2) Lip p ( f s ) + | u αβ | Lip p (cid:0) Φ αβs (cid:1)(cid:3) ( | w | + |∇ w | ) + | w αβ | (cid:12)(cid:12) Φ αβs (cid:12)(cid:12)(cid:9) | v | + nh Lip p ( f y γ ) + | u αβ | Lip p (cid:16) Φ αβy γ (cid:17)i ( | w | + |∇ w | ) + | w αβ | (cid:12)(cid:12)(cid:12) Φ αβy γ (cid:12)(cid:12)(cid:12)o | v γ | + Lip p (cid:0) Φ αβ (cid:1) ( | w | + |∇ w | ) | v αβ | . (4.15)The fundamental theorem of calculus in one variable gives N ( u + v ) − N ( u ) = Z (cid:18) ddt (cid:12)(cid:12) t =0 N ( u + tv ) (cid:19) dt = Z L u + tv v dt . (4.16)Finally, combining this with (4.15) gives that (again at p ) |N ( u + v ) − N ( u ) − L u v | ≤ sup t ∈ [0 , | L u + tv v − L u v |≤ n(cid:12)(cid:12) Φ αβs (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Φ αβy γ (cid:12)(cid:12)(cid:12) + Lip p (cid:0) Φ αβ (cid:1)o ( | v | + |∇ v | ) | Hess v | (4.17) + n Lip p ( f s ) + | u αβ | Lip p (cid:0) Φ αβs (cid:1) + Lip p ( f y γ ) + | u αβ | Lip p (cid:16) Φ αβy γ (cid:17)o ( | v | + |∇ v | ) . (cid:3) Proof of Lemma 4.3.
By Lemma 4.6, M ( u ) is of the form (4.8). Since 0 is a critical pointfor F , we have M (0) = 0. Therefore, Lemma 4.10 gives |M ( u ) − Lu | ≤ C ( | u | + |∇ u | ) + C ( | u | + |∇ u | ) | Hess u | , (4.18)where the constant C = C ( x ) is bounded by C (1 + | x | ) and the constant C is uniformlybounded independent of x ∈ Σ (these bounds follow from Lemma 4.6).Integrating in space (against the Gaussian weight) gives kM ( u ) − Lu k L ≤ C (cid:0) k (1 + | x | ) u k L + k (1 + | x | ) |∇ u | k L (cid:1) + C k ( | u | + |∇ u | ) | Hess u |k L ≤ C (cid:0) k (1 + | x | ) u k L + k (1 + | x | ) |∇ u | k L (cid:1) + C k (1 + | x | ) − | Hess u | k L , (4.19)where the last inequality used the absorbing inequality2 ( | u | + |∇ u | ) | Hess u | ≤ (1 + | x | ) ( | u | + |∇ u | ) + (1 + | x | ) − | Hess u | . (4.20)To get rid of the | x | ’s in the first two terms, we use Lemma 3.4 to get k| x | u k L ≤ C k u k W , ≤ C k u + |∇ u | k L , (4.21) k| x | |∇ u | k L ≤ C k|∇ u | + |∇ R n − k |∇ u || k L . (4.22)To get the first claim, we substitute these bounds back into (4.19) kM ( u ) − Lu k L ≤ C (cid:13)(cid:13) u + |∇ u | + |∇ R n − k |∇ u || + (1 + | x | ) − | Hess u | (cid:13)(cid:13) L . (4.23) To get the second claim, we first use the fundamental theorem of calculus and the definitionof the gradient to get F ( u ) − F ( C k ) − h u, Lu i L = Z ddt (cid:20) F ( tu ) − t h u, Lu i L (cid:21) dt = Z h u, M ( tu ) − t Lu i L dt . (4.24)Since the Cauchy-Schwarz inequality bounds the integrand by k u k L kM ( tu ) − t Lu k L , thesecond claim now follows from the first. (cid:3) The gradient Lojasiewicz inequality.
We will prove Proposition 4.1 and then useit to prove our gradient Lojasiewicz inequality using our first Lojasiewicz inequality.
Proof of Proposition 4.1.
Step 1 : Cutting off to get a compactly supported perturbation ofthe cylinder.
Unlike this proposition, both Lemma 4.3 and the results of the previous sectionare for entire graphs over a cylinder. Thus, we fix a cutoff function ψ with 0 ≤ ψ ≤ B ˜ R − and zero outside of B ˜ R and set u = ψ ˜ u . (4.25)Observe that u has k u k C ≤ C n k ˜ u k C ≤ C n ǫ , where C n depends on the C norm of ψ and,thus, depends only on n . Since ψ is supported in B ˜ R and | ψ | ≤
1, we have k u k L ≤ k ˜ u k L ( B ˜ R ) . (4.26)Finally, using the exponential decay of the Gaussian, we see that | F (Σ) − F (Graph u ) | ≤ C λ ˜ R n − e − ( ˜ R − , (4.27) kM ( u ) k L ≤ C k φ k L ( B ˜ R ) + C n e − ( ˜ R − , (4.28)where φ here is the φ for Σ and C, C n depend only on n . Step 2 : The gradient Lojasiewicz inequality for the compact perturbation.
To simplifynotation, define F ( u ) by F ( u ) = F (Graph u ) − F ( C k ) , (4.29)and, given a function v , set k v k ≡ (cid:13)(cid:13)(cid:13) v + |∇ v | + (cid:12)(cid:12) Hess v ( · , R n − k ) (cid:12)(cid:12) + (1 + | x | ) − | Hess v | (cid:13)(cid:13)(cid:13) L . (4.30)Assuming that k u k C is sufficiently small, then Lemma 4.3 gives C so that(L1) |kM ( u ) k L − k Lu k L | ≤ C k u k .(L2) (cid:12)(cid:12) F ( u ) − h u, Lu i L (cid:12)(cid:12) ≤ C k u k L k u k .Here we also used the Kato inequality |∇ R n − k |∇ v || ≤ (cid:12)(cid:12) Hess v ( · , R n − k ) (cid:12)(cid:12) . (4.31)We will divide into cases depending on the projection of u to the kernel K of L . Let C be the constant from (L1) and (L2). Case 1 : Suppose first that u satisfies k u K k ≤ ǫ k u ⊥ k βW , , (4.32)where ǫ > β ∈ [0 , Lemma 3.22 gives k u k ≤ k u K k + 2 k u ⊥ k ≤ ǫ + C k u k C ) k u ⊥ k W , . (4.33)Using (L1) and (3.13) and then using (4.33) gives kM ( u ) k L ≥ k Lu k L − C k u k ≥ µ k u ⊥ k W , − C k u k ≥ ( µ − C [ ǫ + C k u k C ]) k u ⊥ k W , . (4.34)We now choose ǫ > k u k C so that 2 C [ ǫ + C k u k C ] ≤ µ and, thus, kM ( u ) k L ≥ µ k u ⊥ k W , . (4.35)We will show that F ( u ) is higher order in k u ⊥ k W , . Since L is symmetric and Lu K = 0,Cauchy-Schwarz and the bound on L from W , to L by (3.12) give |h u, Lu i L | = (cid:12)(cid:12) h u ⊥ , Lu ⊥ i L (cid:12)(cid:12) ≤ C k u ⊥ k L k u ⊥ k W , . (4.36)Substituting this into (L2), using that k u k ≤ C k u ⊥ k W , (by (4.33)), and applying thetriangle inequality k u k L ≤ k u K k L + k u ⊥ k L gives | F ( u ) | ≤ C k u ⊥ k L k u ⊥ k W , + C k u k L k u k ≤ C k u ⊥ k L k u ⊥ k W , + C k u K k L k u ⊥ k W , . (4.37)The first term on the right side is trivially bounded by C k u ⊥ k W , . To bound the last term,we use that the cylinder has finite Gaussian area so that k u K k L ≤ C k u K k L ≤ C k u K k , (4.38)to get that k u K k L k u ⊥ k W , ≤ C k u K k k u ⊥ k W , ≤ C k u ⊥ k β W , , (4.39)where the last inequality used (4.32). Putting all of this together (and noting that k u ⊥ k W , is bounded) gives | F ( u ) | ≤ C k u ⊥ k β W , ≤ C kM ( u ) k β L , (4.40)where the last inequality is (4.35). Combining this with the bound on kM ( u ) k L from (4.28)gives | F ( u ) | ≤ C k φ k β L ( B ˜ R ) + C e − (3+ β )( ˜ R − . (4.41) Case 2 : Suppose now that u satisfies k u K k > ǫ k u ⊥ k βW , . (4.42) Note that k u ⊥ k W , is small so k u ⊥ k βW , ≤ k u ⊥ k W , . Lemma 3.22 gives C so that k u ⊥ k ≤ C k u k C k u ⊥ k W , ≤ C k u K k β , (4.43)where the last inequality is (4.42). Using the squared triangle inequality and (4.43) gives k u k ≤ k u K k + 2 k u ⊥ k ≤ C k u K k β , (4.44)where the last inequality uses that k u K k is bounded. Using (L2) and (3.12), then (4.42)and (4.44) (and the projection inequality k u ⊥ k L ≤ k u k L ), we get | F ( u ) | ≤ C k u ⊥ k L k u ⊥ k W , + C k u k L k u k ≤ C k u k L k u K k β . (4.45)However, since Lemma 3.22 and the projection inequality k u K k L ≤ k u k L give that k u K k ≤ C K k u K k L ≤ C K k u k L , (4.46)we conclude that | F ( u ) | ≤ C k u k β β L ≤ C k ˜ u k β β L ( B ˜ R ) , (4.47)where the last inequality is the Gaussian L bound on u from (4.26).If we now combine the bounds from the two cases, then we see that | F ( u ) | ≤ C k φ k β L ( B ˜ R ) + C e − (3+ β )( ˜ R − + C k ˜ u k β β L ( B ˜ R ) . (4.48)Finally, we use the triangle inequality to combine this with the bound (4.27) on the F functional from Step 2 and Step 3 to complete the proof. (cid:3) We will use the following elementary lemma to control graphical bounds when we write asurface as a graph over two nearby cylinders.
Lemma 4.49.
There exists ǫ = ǫ ( n ) > , Σ ∈ C k , 5 √ n ≤ R < R and • B R ∩ Σ is the graph of u over Σ with | u | + |∇ u | ≤ ǫ , • B R ∩ Σ is the graph of u over Σ with k u k C ,α ≤ ǫ ,then we get for R = min { R , R } that • B R ∩ Σ is the graph of u over Σ with k u k C ≤ ¯ ǫ . Proof.
Since B R ∩ Σ is ǫ C -close to Σ and ǫ close to Σ , we get that the distance betweenΣ and Σ in B R is at most 2 ǫ . Since the distance between cylinders grows linearly in theradius, we conclude that the distance between Σ and Σ in B R is at most 4 ǫ . The lemmafollows easily from this. (cid:3) Proof of Theorem 0.26.
The result will follow by combining the L closeness to a cylindergiven by the first Lojasiewicz inequality and Proposition 4.1. Note that we can assume that R is large and k φ k L ( B R ) is small since the inequality is otherwise trivially true. Step 1 : Fixing the nearby cylinder.
The Lojasiewicz inequality of Theorem 0.24 gives acylinder Σ k ∈ C k so that B ˜ R ∩ Σ is the graph of ˜ u over Σ k with k ˜ u k C ≤ ǫ , where b ℓ,n ∈ (0 , satisfies lim ℓ →∞ b ℓ,n = 1, and ˜ R = max (cid:26) r ≤ R (cid:12)(cid:12) R n +5 (cid:26) e − b ℓ,n R + k φ k bℓ,n L ( B R ) (cid:27) e r ≤ ˜ C (cid:27) , (4.50)where ˜ C depends on n, λ , ℓ, C ℓ . Combining this with Lemma 4.49, we extend ˜ u out to¯ R = min { R, R } so that( ⋆ ) B ¯ R ∩ Σ is the graph of ˜ u over Σ k with k ˜ u k C ≤ ¯ ǫ ,( ⋆ ) k ˜ u k L ( B ¯ R ) ≤ C R ρ (cid:26) k φ k b ℓ,n L ( B R ) + e − bℓ,n R (cid:27) ,where C = C ( n, ℓ, C ℓ , λ ) and ρ = ρ ( n ). Step 2 : Using the first Lojasiewicz to get the second . Proposition 4.1 gives | F (Σ) − F ( C k ) | ≤ C (cid:26) k φ k β L ( B ¯ R ) + (1 + ¯ R n − ) e − (3+ β )( ¯ R − + k ˜ u k β β L ( B ¯ R ) (cid:27) , (4.51)where C = C ( n, λ ). To bound the last term in (4.51), we use ( ⋆ ) to get k ˜ u k β β L ( B ¯ R ) ≤ C R β β ρ (cid:26) k φ k b ℓ,n β β L ( B R ) + e − bℓ,n (3+ β ) R β ) (cid:27) . (4.52)To deal with the exponential term in (4.51), we consider two cases. Suppose first that¯ R < R , so that ¯ R = 2 ˜ R and the definition of ˜ R gives R n +5 (cid:26) e − b ℓ,n R + k φ k bℓ,n L ( B R ) (cid:27) e ˜ R = ˜ C . (4.53)Since ¯ R = 2 ˜ R in this case, we havee − ¯ R = h e − ˜ R i ≤ C R n +20 n e − b ℓ,n R + k φ k b ℓ,n L ( B R ) o . (4.54)We can assume that ¯ R > (cid:16) ¯ R − R (cid:17) > /
2. Raising (4.54) to the β (cid:16) ¯ R − R (cid:17) > β power, we bound the exponential term in (4.51) by a constant times a power of R timese − b ℓ,n (3+ β ) R + k φ k b ℓ,n (3+ β )2 L ( B R ) ≤ e − (3+ β )( R − + k φ k b ℓ,n (3+ β )2 L ( B R ) , (4.55)where the last inequality used also that b ℓ,n is close to one, and in particular at least 1 / R < R , but it also obviously holds in the casewhen when ¯ R = R (and the φ term is unnecessary).Putting it all together, | F (Σ) − F ( C k ) | is bounded by C R ρ ′ times k φ k β L ( B ¯ R ) + (cid:26) k φ k b ℓ,n β β L ( B R ) + e − bℓ,n (3+ β ) R β ) (cid:27) + (cid:26) e − (3+ β )( R − + k φ k b ℓ,n (3+ β )2 L ( B R ) (cid:27) , where we have grouped terms together based on where they came from in (4.51). Finally,the first and fifth terms can be absorbed in the second term. (cid:3) We choose ˜ R to make k ˜ u k C small by (2.51). Compatibility of the shrinker and cylindrical scales
One of the main difficulties in this paper is that the singularities are not compact and,thus, surfaces cannot generally be written as entire graphs over a cylinder. As a result, ourestimates include “error terms” coming from cut off functions. Thus, a surface is close tothe cylinder if a large part of it can be written as a small graph over the cylinder.Given a hypersurface Σ ⊂ R n +1 , we will prove a lower bound for the scale on which it is“roughly cylindrical” in Theorem 5.3 below. This essentially bounds the error terms in ourLojasiewicz inequalities by a power greater than one of |∇ Σ F | , which is crucial in the nextsection when we prove uniqueness of tangent flows. It will also imply that the size of thegraphical region is growing at a definite rate under the rescaled MCF.5.1. The cylindrical scale and the shrinker scale.
Recall that the cylindrical scale r ℓ (Σ)is the largest radius where Σ can be written as a small C ,α graph over a cylinder with auniform bound on ∇ ℓ A . Namely, given a fixed ǫ >
0, an integer ℓ and a constant C ℓ , r ℓ (Σ)is the maximal radius where • B r ℓ (Σ) ∩ Σ is the graph over a cylinder in C k of a function u with k u k C ,α ≤ ǫ and |∇ ℓ A | ≤ C ℓ .The constant ǫ is fixed, but we have yet to choose ℓ and C ℓ . (The constant ℓ will be chosenlarge to get good bounds on lower derivatives by interpolation and then C ℓ will be chosen.)The point of this section is to prove that these cylindrical scales are large enough that theerror terms in our Lojasiewicz inequalities can be absorbed. The scale R that we have tobeat is roughly given by e − R = |∇ Σ F | . Thus, we define a “shrinker scale” R (Σ) bye − R = |∇ Σ F | , (5.1)with the convention that R (Σ) is infinite when Σ is a complete shrinker. When Σ t flows bythe rescaled MCF, we define the shrinker scale (also denoted by R (Σ t )) to bee − R t )2 = Z t +1 t − |∇ Σ s F | ds = F (Σ t − ) − F (Σ t +1 ) . (5.2)The main result of this section is the following theorem that shows that the cylindricalscale is a fixed factor larger than the shrinker scale : Theorem 5.3.
There exist µ > C so that if Σ t flows by the rescaled MCF and λ (Σ t ) ≤ λ , then, given any ℓ , there exists C ℓ (depending on ℓ ) so that(1 + µ ) R (Σ t ) ≤ min t − / ≤ s ≤ t +1 r ℓ (Σ s ) + C . (5.4)To understand this, observe that Theorem 2.54 gives uniform graphical estimates on anyscale less than R (Σ). To apply Theorem 2.54, we need uniform curvature bounds and a lowerbound for H on this larger scale. We will establish these uniform bounds on the larger scaleby an extension and improvement argument, where Theorem 2.54 gives uniform bounds onlarger scales (this is the improvement part). Roughly speaking, the extension argument will To see this, note that the larger exponentially decaying term on the right in Theorem 0.26 is essentiallye − R . We need to bound this by a power greater than one of |∇ Σ F | = k φ k L . use curvature estimates for MCF to get bounds on a larger scale forward in time, then usethe bounds on φ to pull these bounds backwards in time under the rescaled MCF. Repeatingthis gets us as close as we want to the scale R (Σ) and gets us uniform curvature bounds on alarger scale than R (Σ). The final step is to get graphical estimates on a larger scale too; forthis, we cannot use Theorem 2.54. Rather, we get these graphical estimates from estimatesfor MCF and a scaling argument to relate MCF and rescaled MCF. Backward curvature estimates.
Recall that, when Σ ⊂ R n +1 is a hypersurface, φ is defined to be φ = h x, n i − H , so that Σ t flows by the rescaled MCF if ∂ t x = φ n and |∇ Σ F | = k φ k L ≡ Z Σ φ e − | x | . (5.5)In this subsection, we will show the following curvature estimate for the rescaled MCF: Proposition 5.6.
Given n , λ , there exists s ≥ δ > / ≥ τ >
0, there exists µ >
0, such that if Σ t flows by the rescaled MCF, λ ≥ λ (Σ t ), t ≥ t + τ , x ∈ B R − s and Z t t Z B R +2 ∩ Σ t φ e − | x | ≤ µ e − ( R +2)24 R ( t − t + 1) , (5.7) sup B s √ τ ( x ) ∩ Σ t | A | ≤ δ τ − , (5.8)then for all t ∈ [ t − log(1 − τ / , t − log(1 − τ )] and any ℓ sup B √ τ (cid:16) e
12 ( t − t x (cid:17) ∩ Σ t n | A | + τ ℓ (cid:12)(cid:12) ∇ ℓ A (cid:12)(cid:12) o ≤ C ℓ τ , (5.9)where C ℓ depends on n and ℓ .We will need the following elementary lemma: Lemma 5.10.
If Σ t flows by the rescaled MCF and 0 ≤ η ≤ R n +1 , then for t ≤ t and τ > Z Σ t η e − | x − x | τ − Z Σ t η e − | x − x | τ = Z t t Z Σ t h∇ η, n i φ e − | x − x | τ + Z t t Z Σ t h x , n i τ φ η e − | x − x | τ (5.11) + (cid:18) − τ (cid:19) Z t t Z Σ t h x, n i φ η e − | x − x | τ − Z t t Z Σ t φ e − | x − x | τ . This final step can not be iterated since it has a loss in the estimates and we can no longer applyTheorem 2.54 to get rid of the loss. Proof. If f ∈ R n +1 → R is a smooth function with compact support, then ddt Z Σ t f e − | x | = Z Σ t h∇ f, n i φ e − | x | − Z Σ t f φ e − | x | = Z Σ t h∇ log f, n i φ f e − | x | − Z Σ t φ f e − | x | . (5.12)If we set f ( x ) = η e | x | e − | x − x | τ , then ∇ log f = ∇ log η + x − x − x τ = ∇ log η + x τ + 12 (cid:18) − τ (cid:19) x . (5.13)Therefore ddt Z Σ t η e − | x − x | τ = Z Σ t h∇ η, n i φ e − | x − x | τ + Z Σ t h x , n i τ φ η e − | x − x | τ + (cid:18) − τ (cid:19) Z Σ t h x, n i φ η e − | x − x | τ − Z Σ t φ η e − | x − x | τ . (5.14)The lemma now follows by integrating from t to t . (cid:3) Corollary 5.15.
Given ǫ >
0, 1 ≥ τ >
0, and λ , there exists µ = µ ( ǫ, τ, λ ) > s = s ( ǫ, λ ) ≥ t flows by the rescaled MCF, λ ≥ λ (Σ t ), x ∈ B R − s , and t > t Z t t Z B R +2 ∩ Σ t φ e − | x | ≤ µ e − ( R +2)24 R ( t − t + 1) , (5.16) (4 πτ ) − n Z Σ t e − | x − x | τ ≤ ǫ , (5.17)then (4 πτ ) − n Z Σ t e − | x − x | τ ≤ ǫ . (5.18) Proof.
Observe first that by the entropy bound λ (Σ t ) ≤ λ , there exists s > y ∈ R n +1 and all t (4 πτ ) − n Z Σ t \ B s √ τ ( y ) e − | x − y | τ ≤ ǫ . (5.19)If we choose a non-negative function η with η ≤ |∇ η | ≤ η = 1 on B R , and η = 0 outside B R +2 , then Lemma 5.10 gives Z B R ∩ Σ t e − | x − x | τ ≤ Z Σ t η e − | x − x | τ ≤ Z B R +2 ∩ Σ t e − | x − x | τ + Z t t Z ( B R +2 \ B R ) ∩ Σ t | φ | e − | x − x | τ + (cid:18) τ − (cid:19) Z t t Z B R +2 ∩ Σ t |h x, n i| | φ | e − | x − x | τ + Z t t Z B R +2 ∩ Σ t |h x , n i| τ | φ | e − | x − x | τ + Z t t Z B R +2 ∩ Σ t φ e − | x − x | τ . (5.20) Combining the terms that are linear in φ gives Z B R ∩ Σ t e − | x − x | τ ≤ Z B R +2 ∩ Σ t e − | x − x | τ + | x | τ + (cid:0) τ − (cid:1) ( R + 2)2 ! Z t t Z B R +2 ∩ Σ t | φ | e − | x − x | τ + Z t t Z B R +2 ∩ Σ t φ ≤ Z B R +2 ∩ Σ t e − | x − x | τ + R + 2 τ Z t t Z B R +2 ∩ Σ t | φ | e − | x − x | τ + Z t t Z B R +2 ∩ Σ t φ . (5.21)By the entropy bound λ (Σ t ) ≤ λ and the Cauchy-Schwarz inequality, we have Z t t Z B R +2 ∩ Σ t | φ | e − | x − x | τ ≤ Z t t Z B R +2 ∩ Σ t φ e − | x − x | τ ! q (4 πτ ) n ( t − t ) λ ≤ q (4 πτ ) n ( t − t ) λ Z t t Z B R +2 ∩ Σ t φ ! (5.22) ≤ q (4 πτ ) n ( t − t ) λ e ( R +2)28 Z t t Z B R +2 ∩ Σ t φ e − | x | ! . We can therefore bound the two last terms in (5.21) as follows R + 2 τ Z t t Z B R +2 ∩ Σ t | φ | e − | x − x | τ + Z t t Z B R +2 ∩ Σ t φ ≤ C R + 2 τ q (4 πτ ) n ( t − t ) e ( R +2)28 Z t t Z B R +2 ∩ Σ t φ e − | x | ! + e ( R +2)24 Z t t Z B R +2 ∩ Σ t φ e − | x | ≤ C ( µ/τ + µ ) . (5.23)Using (5.19), (5.21), and (5.23) we get that(4 πτ ) − n Z Σ t e − | x − x | τ = (4 πτ ) − n Z B R ∩ Σ t e − | x − x | τ + (4 πτ ) − n Z Σ t \ B R e − | x − x | τ ≤ (4 πτ ) − n Z Σ t e − | x − x | τ + C ( µ/τ + µ ) + ǫ . (5.24)Choosing µ sufficiently small gives the corollary. (cid:3) We will apply this corollary in combination with Brakke’s regularity result to get curva-ture estimates at an earlier time-slice in terms of curvature estimates at a later time-slice.By White’s [W3] version of Brakke’s regularity result [B], there exist constants ǫ and C B depending on n and λ such that if M s ⊂ R n +1 flow ( s <
0) by the MCF, λ ( M s ) ≤ λ , andfor some s < − πs ) − n Z M s e | x − x | s ≤ ǫ , (5.25) then for all s ∈ [ − s ,
0] sup M s ∩ B √ − s ( x ) | A | ≤ C B − s . (5.26)We can use the correspondence between MCF and rescaled MCF to translate this into asimilar curvature estimate for rescaled MCF. Namely, if Σ t is a rescaled MCF with entropyat most λ and there is some τ ∈ (0 , /
2) so that(4 πτ ) − n Z Σ t e − | x − x | τ ≤ ǫ , (5.27)then for all t ∈ [ t − log(1 − τ / , t − log(1 − τ )] we havesup Σ t ∩ B √ τ (cid:16) e
12 ( t − t x (cid:17) | A | ≤ C B τ . (5.28)This is proven by writing the rescaled flow Σ t as e ( t − t ) M s where where s = 1 − e t − t − τ and M s is the MCF with M − τ = Σ t . (Here we have used that the result of Brakke/White isuniform in Σ or more precise uniform in the point x where it is centered as for the rescaledMCF when the point x is fixed this mean that the original “fixed” point x for the MCFevolves by e ( t − t ) x .) Proof of Proposition 5.6.
Combining the above consequence of Brakke’s theorem with Corol-lary 5.15 gives the | A | bound in Proposition 5.6 for t in the time interval[ t − log(1 − τ / , t − log(1 − τ )] . (5.29)The bounds on higher derivatives of A then follow from this and the interior estimates ofEcker and Huisken, [EH]. (cid:3) A mean value inequality.
In the next lemma, we will use that if Σ t flow by therescaled MCF, then (see section 2 of [CIMW])( ∂ t − L ) φ = 0 where L = L + | A | + 12 . (5.30)Hence, ( ∂ t − L ) φ = 2 φ ( ∂ t − L ) φ − |∇ φ | = φ (2 | A | + 1) − |∇ φ | . (5.31) Lemma 5.32.
There exists a constant C so that if Σ t flow by the rescaled MCF for t ∈ [ t , t ], r + 1 ≤ min t ≤ s ≤ t r (Σ s ) and 0 < β < ( t − t ) /
2, thenmax s ∈ [ t + β,t ] |∇ Σ s F | B r ≤ ( C + 1 /β ) ( F (Σ t ) − F (Σ t )) . (5.33) Z t t + β Z B r ∩ Σ s |∇ φ | e − | x | ≤ ( C + 1 /β ) ( F (Σ t ) − F (Σ t )) . (5.34) Proof.
Fix a compactly supported function η on R n +1 with 1 ≤ η ≤ η identically one on B r , η vanishes outside B r +1 , and |∇ η | ≤
2. If we restrict η to Σ t , then the flow equation and(5.31) give(5.35) ∂ t (cid:0) φ η (cid:1) = ( η ) t φ + η ∂ t φ = φ h∇ η , n i + η (cid:0) L φ + φ (2 | A | + 1) − |∇ φ | (cid:1) . Using this and the equation for the derivative of the weighted measure, and integrating byparts to take L off of φ , we get ∂ t (cid:18)Z Σ t φ η e − | x | (cid:19) = − Z Σ t φ η e − | x | + Z Σ t (cid:0) | A | + 1 (cid:1) φ η e − | x | − Z Σ t |∇ φ | η e − | x | + Z Σ t φ h∇ η , n i e − | x | − Z Σ t h∇ φ , ∇ η i e − | x | . (5.36)Using the absorbing inequalities 2 φ η |∇ η | ≤ φ η + φ |∇ η | and 4 η | φ ||∇ η ||∇ φ | ≤ η |∇ φ | +4 φ |∇ η | , we get ∂ t (cid:18)Z Σ t φ η e − | x | (cid:19) ≤ Z Σ t (cid:8)(cid:0) | A | + 1 (cid:1) η + 5 |∇ η | (cid:9) φ e − | x | − Z Σ t |∇ φ | η e − | x | ≤ C Z Σ t φ e − | x | − Z Σ t |∇ φ | η e − | x | . (5.37)Suppose that s ∈ [ t + β, t ]. To prove (5.33), we integrate (5.37) to get Z Σ s φ η e − | x | ≤ min [ t ,t + β ] Z Σ t φ η e − | x | + C Z st Z Σ t φ e − | x | ≤ ( C + 1 /β ) Z st Z Σ t φ e − | x | ≤ ( C + 1 /β ) ( F (Σ t ) − F (Σ t )) . (5.38)Finally, to get (5.34), we integrate (5.37) from t + β to t and use (5.33) to bound thecontributions at the end points. (cid:3) Uniform short time stability of the cylinder.
The last result that we will need forproving Theorem 5.3 is the following elementary short time uniform stability of the cylinderunder MCF with bounded curvature:
Lemma 5.39.
Given
R > √ n , ǫ > C , there exist δ > θ > M t isa MCF with(1) B R +2 ∩ M − is a C ,α graph over Σ ∈ C k with norm at most δ .(2) | A | + |∇ A | + |∇ A | + |∇ A | ≤ C on B R +2 ∩ M t for t ∈ [ − − /C , − /C ].Then for each t ∈ [ − , θ −
1] we have that • B R ∩ M t is a C ,α graph over √− t Σ with norm at most ǫ . Proof.
Since | A | is bounded, the MCF equation implies that | ∂ t x | is also bounded. Likewise,the bound on |∇ A | (and thus on |∇ H | ) and the evolution equation for the normal (see lemma7 . | ∂ t n | is also uniformly bounded. Combining these two bounds, itfollows that B R +1 ∩ M t remains a graph over Σ of a function u with a uniform bound | ∂ t u | + | ∂ t ∇ u | ≤ C for t ∈ [ − − θ , − θ ] , (5.40)where θ > C depend on C , ǫ, n . Similarly, the higher derivative bounds on A thenyield bounds on higher derivatives of u and the lemma follows immediately. (cid:3) Proof of Theorem 5.3.
We are now prepared to prove Theorem 5.3 which shows thatthe cylindrical scale is a fixed factor larger than the shrinker scale.
Proof. (of Theorem 5.3). The theorem follows by an extension and improvement argumentthat is inspired by a similar argument for shrinkers in [CIM]. (1) Extending the scale.
Given ℓ , we will show that there exist δ >
0, ¯ s > θ > R , C and C ℓ so that if(A1) B R ∩ Σ s is a graph of u over some Σ ∈ C k with k u k C ,α ≤ δ for each s ∈ [ t − ¯ s, t + ¯ s ]for some R ∈ [ R , R (Σ t )] and t ∈ [ t − / , t + 1 − ¯ s ]then, for every s ∈ [ t − ¯ s, t + ¯ s ], we have(A2) r ℓ (Σ t ) ≥ (1 + θ ) R and |∇ Σ s F | B (1+ θ ) R ≤ C ( F (Σ t − ) − F (Σ t +1 )).The key observation is that the cylindrical estimates and global entropy bound implythat the local Gaussian densities on some fixed scale are almost one. Thus, White’s Brakkeestimate [W3] gives a curvature bound on a larger region B (1+ κ ) R with κ > while only coming in by a fixed additive amount . As long as R is sufficiently large, themultiplicative gain beats the additive loss and, thus, the bound on A extends to a larger scalewith no loss in time. Ecker-Huisken [EH] then gives uniform higher derivative bounds on A .We can now apply Lemma 5.39 on a unit scale but centered at points out to the extendedscale to get the cylindrical estimates on the larger scale. Finally, using the curvature bounds,Lemma 5.32 gives the constant C so that |∇ Σ s F | B (1+ θ ) R ≤ C ( F (Σ t − ) − F (Σ t +1 )). (2) The improvement below the shrinker scale . The Lojasiewicz inequality of Theorem2.54 will give an improved bound on the larger scale if we are below the shrinker scale:Given τ > δ > C , ℓ and C ℓ , there exist ℓ and R so that if ℓ ≥ ℓ and R ∈ [ R , R (Σ t )]satisfies R ≤ r ℓ (Σ s ) and |∇ Σ s F | B R ≤ C ( F (Σ t − ) − F (Σ t +1 )) , (5.41)then B (1 − τ ) R ∩ Σ s is a graph of u over some Σ ∈ C k with k u k C ,α ≤ δ . Putting it together : The point is to choose τ much smaller than θ , so that the gain inscale from extending in (1) beats the loss in scale from the improvement in (2). We can thenapply the two steps iteratively to get a fixed factor greater than one beyond the shrinkerscale, giving the theorem. (cid:3) The gradient Lojasiewicz inequality and uniqueness
In this section, we will use the gradient Lojasiewicz inequality of Theorem 0.26 and thecompatibility of the shrinker and cylindrical scales of the previous section to prove a gradientLojasiewicz inequality for rescaled MCF. We will show that this inequality implies uniquenessof the tangent flow at a cylindrical singularity, thus completing the proof of Theorem 0.2.6.1.
Mean value inequalities.
In this subsection, we will prove a mean value inequalitythat is needed for the gradient Lojasiewicz inequality. The argument follows that of Lemma5.32 in the previous section with the gradient of φ in place of φ essentially using the equationthat one gets from taking the derivative of equation (5.30) to get the following: Lemma 6.1.
There exists a constant C so that if Σ t flow by the rescaled MCF, and r ≤ min t − / ≤ s ≤ t +1 r ℓ (Σ s ), thenmax s ∈ [ t − ,t +1] Z B r ∩ Σ t |∇ φ | e − | x | ≤ C (1 + r ) ( F (Σ t − ) − F (Σ t +1 )) . (6.2) Z t +1 t − Z B r ∩ Σ s | Hess φ | e − | x | ≤ C (1 + r ) ( F (Σ t − ) − F (Σ t +1 )) . (6.3)Lemma 6.1 will follow from the same argument as in the proof of Lemma 5.32 (togetherwith the result of Lemma 5.32) provided we have the following: Lemma 6.4.
If Σ t flow by the rescaled MCF, then( ∂ t − L ) |∇ φ | ≤ − | Hess φ | + C |∇ φ | + φ , (6.5)where C depends only on n and the bounds for A and ∇ A . Proof.
To prove this, note first that if Σ ⊂ R n +1 is a hypersurface, f : R n +1 → R is asmooth function, X , Y ∈ T x Σ, thenHess R n +1 f ( X, Y ) = h∇ X (cid:0) ∇ T f + h∇ f, n i n (cid:1) , Y i = Hess Σ f ( X, Y ) − h∇ f, n i A ( X, Y ) . (6.6)Recall also that if Σ is a manifold (not necessarily embedded in Euclidean space), then theBochner formula for the drift Laplacian ∆ f u = ∆ u − h∇ f, ∇ u i is12 ∆ f |∇ u | = | Hess u | + h∇ ∆ f u, ∇ u i + Ric f ( ∇ u, ∇ u ) . (6.7)Here Ric f = Ric + Hess f is the Bakry- ´Emery Ricci curvature.We will use that if Σ t ⊂ R n +1 is a one-parameter family of hypersurfaces moving by therescaled MCF and u = u ( x, t ) : Σ t × R → R is a smooth function, then ∂ t |∇ T u | = 2 h∇ T ∂ t u, ∇ T u i + 2 (cid:18) h x, n i − H (cid:19) A ( ∇ T u, ∇ T u ) . (6.8) Proof of (6.8) : To see this, extend u to a function on R n +1 × R so that on Σ t ∇ u = ∇ T u and ∂ t u = h∇ u, ∂ t x i + u t = u t , where u t is the t derivative of u as a function on R n +1 × R and the rescaled MCF equation is ∂ t x = (cid:18) h x, n i − H (cid:19) n . (6.9)Therefore, differentiating ∇ u = ∇ u ( x ( t ) , t ) on Σ t , the chain rule gives ∂ t ∇ u = ∇ ∂ t x ∇ u + ∇ u t = (cid:18) h x, n i − H (cid:19) ∇ n ∇ u + ∇ ∂ t u . (6.10)Using this, the symmetry of the Hessian, and the definition of A gives12 ∂ t |∇ T u | = 12 ∂ t |∇ u | = h ∂ t ∇ u, ∇ u i = (cid:18) h x, n i − H (cid:19) h∇ ∇ u ∇ u, n i + h∇ ∂ t u, ∇ u i = (cid:18) h x, n i − H (cid:19) A ( ∇ u, ∇ u ) + h∇ ∂ t u, ∇ u i , (6.11)completing the proof of (6.8). Let f = | x | so that (6.6) givesRic f ( ∇ u, ∇ u ) = Ric( ∇ u, ∇ u ) + |∇ u | h x, n i A ( ∇ u, ∇ u ) . (6.12)Therefore, combining (6.8) and the Bochner formula (6.7) gives( ∂ t − L ) |∇ u | = − | Hess u | + 2 h∇ ( ∂ t − L ) u, ∇ u i− ∇ u, ∇ u ) − |∇ u | − H A ( ∇ u, ∇ u ) . (6.13)Suppose now that u = φ , so that ( ∂ t − L ) φ = ( | A | + ) φ by (5.30). Finally, (6.5) followsfrom (6.13) and the absorbing inequality 2 | φ h∇| A | , ∇ φ i| ≤ φ + C |∇ φ | . (cid:3) A discrete gradient Lojasiewicz inequality for rescaled MCF.
The next theoremgives a discrete version of a gradient Lojasiewicz inequality for rescaled MCF.
Theorem 6.14.
Given n and λ , there exist constants K, ¯ R, ǫ and τ ∈ (1 / ,
1) so that if Σ s is a rescaled MCF for s ∈ [ t − , t + 1] satisfying • λ (Σ s ) ≤ λ . • B ¯ R ∩ Σ s is a C ,α graph over some cylinder in C k with norm at most ǫ for each s .Then we have ( F (Σ t ) − F ( C )) τ ≤ K ( F (Σ t − ) − F (Σ t +1 )) . (6.15) Proof.
Given any β ∈ [0 ,
1) and R ∈ [1 , r ℓ (Σ t ) − | F (Σ t ) − F ( C ) | ≤ C R ρ (cid:26) k φ k c ℓ,n β β L ( B R ∩ Σ t ) + e − R c ℓ,n + e − R ( β ) (cid:27) , (6.16)where C = C ( n, ℓ, C ℓ , λ ), ρ = ρ ( n ) and c ℓ,n ∈ (0 ,
1) satisfies lim ℓ →∞ c ℓ,n = 1. We will boundeach term by a power greater than 1 / F (Σ t − ) − F (Σ t +1 )).We defined the shrinker scale R (Σ t ) in (5.2) bye − R t )2 = Z t +1 t − |∇ Σ s F | ds = F (Σ t − ) − F (Σ t +1 ) . (6.17)If we set R + 2 ≡ min t − / ≤ s ≤ t +1 r ℓ (Σ s ), then Theorem 5.3 gives µ > C so that R ≥ (1 + µ ) R (Σ t ) − C , (6.18)as long as we are willing to choose C ℓ sufficiently large depending on ℓ . The crucial point isthat µ does not change when we take ℓ larger, although C ℓ does depend on ℓ .Lemmas 5.32 gives a constant C so that k φ k L ( B R ∩ Σ t ) ≤ C ( F (Σ t − ) − F (Σ t +1 )) . (6.19)We first choose β ∈ [0 ,
1) so that(1 + µ ) (cid:18) β (cid:19) > . (6.20)This takes care of the third term in (6.16). Now we choose ℓ large so that c ℓ,n (cid:18) β β (cid:19) > µ ) c ℓ,n > . (6.21) This takes care of the first two terms. Once we choose ℓ , then Theorem 5.3 gives C ℓ and,thus, determines the multiplicative factor K . (cid:3) An extension of “Lojasiewicz theorem”.
Lojasiewicz used the gradient Lojasiewiczinequality to prove convergence of flow lines for the negative gradient flow of an analyticfunction f . We will prove an analogous convergence result where the differential inequality f β ( t ) ≤ − f ′ ( t ) (that follows from the gradient Lojasiewicz) is replaced by the discreteinequality f β ( t ) ≤ f ( t − − f ( t + 1). This assumption is exactly what comes out of ouranalog of the gradient Lojasiewicz inequality, i.e., out of Theorem 0.26.The extension will rely on the following elementary lemma: Lemma 6.22. If f : [0 , ∞ ) → [0 , ∞ ) is a non-increasing function, ǫ, K > t ≥ K f ǫ ( t ) ≤ f ( t − − f ( t + 1) , (6.23)then there exists a constant C such that f ( t ) ≤ C t − ǫ . (6.24)Moreover, if ǫ <
1, then ∞ X j =1 ( f ( j ) − f ( j + 1)) < ∞ . (6.25) Proof.
After replacing f by f /C for some positive constant C , we can assume without lossof generality that 0 < f (0) ≤ K = 1. Set t = 4 2 ǫ f − ǫ (0) /ǫ + 2 and C = f (0) t ǫ , then f (0) = C t − ǫ and hence (6.24) holds for all t ≤ t . Next note that by assumption for all t ≥ f ǫ ( t ) ≤ f ǫ ( t − ≤ f ( t − − f ( t ) . (6.26)Or, equivalently, for all t ≥ f ( t − ≥ f ( t ) (1 + f ǫ ( t )) . (6.27)We would like to show that (6.24) holds; so suppose not and let t be a t where inequality(6.24) fails. After possibly replacing t by t − t , but holds for t −
2. From the choice of C it follows that t > t ≥ f ( t − ≥ f ( t ) (1 + f ǫ ( t )) > C t − ǫ (1 + C ǫ t − ) . (6.28)Combining this with the elementary inequality that (1 + h ) − ǫ ≤ − − − ǫ ǫ h for all h ≤ C ǫ t − = f ǫ (0) ≤ − − ǫ ǫ C ǫ = 2 − − ǫ ǫ f ǫ (0) t ≥ f − ǫ ( t − < C − ǫ t (1 + C ǫ t − ) − ǫ ≤ C − ǫ ( t − − − ǫ ǫ C ǫ ) ≤ C − ǫ ( t − . (6.29)Contradicting that (6.24) holds for t − ǫ < p ∈ (1 , /ǫ ). Cauchy-Schwarz gives that " ∞ X j =1 ( f ( j ) − f ( j + 1)) ≤ " ∞ X j =1 ( f ( j ) − f ( j + 1)) j p ∞ X j =1 j − p . (6.30) The last term is finite since p >
1, so it suffices to prove that ( f ( j ) − f ( j + 1)) j p is summable.However, this follows from the summation by parts formula n X j =1 b j ( a j +1 − a j ) = [ b n +1 a n +1 − b a ] − n − X j =1 a j +1 ( b j +1 − b j )(6.31)with a j = f ( j ) and b j = j p since the decay (6.24) implies that( n + 1) p f ( n + 1) ≤ C ( n + 1) − ǫ ( n + 1) p → , (6.32) ∞ X j =1 f ( j + 1) [( j + 1) p − j p ] ≤ C p ∞ X j =1 ( j + 1) − ǫ + p − < ∞ . (6.33)The first inequality in the second line used that [( j + 1) p − j p ] ≤ p ( j + 1) p − . (cid:3) Uniqueness of tangent flows.
We are now prepared to prove the uniqueness of cylin-drical tangent flows.
Proof of Theorem 0.2.
Let Σ t be the rescaled MCF associated to the cylindrical singularity.It follows from the uniqueness theorem of [CIM] that if a sequence t j → ∞ , then there isa subsequence t ′ j → ∞ so that Σ t ′ j converges with multiplicity one to a cylinder Σ ∈ C k .It follows from White’s Brakke-type theorem, [W3], that this convergence is smooth oncompact subsets. A priori, different sequences could lead to different cylinders (i.e., differentrotations of the same cylinder); the point of this theorem is that this does not occur.Given any fixed large ρ and small ǫ >
0, it follows from the previous paragraph that theremust be some T so that • For each t ≥ T , there is a cylinder in C k so that, for each s ∈ [ t − , t + 1], B ρ ∩ Σ s isa C ,α graph over this cylinder with norm at most ǫ .Therefore, we can apply Theorem 6.14 to Σ t for t ≥ T to get K and µ ∈ (1 / ,
1) so that( F (Σ t ) − F ( C )) µ ≤ K ( F (Σ t − ) − F (Σ t +1 )) . (6.34)This “discrete differential inequality” allows us to apply Lemma 6.22 to conclude that ∞ X j =1 ( F (Σ j ) − F (Σ j +1 ) < ∞ . (6.35)Using Cauchy-Schwarz and that rescaled MCF is the negative gradient flow for F , we have Z ∞ k φ k L (Σ t ) dt ≤ ∞ X j =1 (cid:18) F (Σ j ) Z j +1 j k φ k L (Σ t ) dt (cid:19) ≤ p F (Σ ) ∞ X j =1 ( F (Σ j ) − F (Σ j +1 ) < ∞ , (6.36)where the last inequality is (6.35) and the L and L norms are all weighted Gaussian norms.The uniqueness now follows immediately from Lemma A.48. (cid:3) Appendix A. Geometric quantities on a graph
In this appendix, we will prove some technical results for the geometry of normal expo-nential graphs over a hypersurface. As one consequence, we will prove Lemma 4.6 whichcomputes the gradient of the F functional on graphs over cylinders.Throughout this appendix, Σ u will denote the graph of a function u over a fixed hyper-surface Σ (in most applications Σ will be a cylinder), where Σ u is given by(A.1) Σ u = { x + u ( x ) n ( x ) | x ∈ Σ } . We will assume that | u | is small so Σ u is contained in a tubular neighborhood of Σ wherethe normal exponential map is invertible. Let e n +1 be the gradient of the (signed) distancefunction to Σ; note that e n +1 equals n on Σ.The geometric quantities that we need to compute on Σ u are: • The relative area element ν u ( p ) = q det g uij ( p ) / p det g ij ( p ), where g ij ( p ) is the metricfor Σ at p and g uij ( p ) is the pull-back metric from the graph of u at ( p + u ( p ) n ( p )). • The mean curvature H u ( p ) of Σ u at ( p + u ( p ) n ( p )). • The support function η u ( p ) = h p + u ( p ) n ( p ) , n u i , where n u is the normal to Σ u . • The speed function w u ( p ) = h e n +1 , n u i − evaluated at ( p + u ( p ) n ( p )).The mean curvature and the support function directly appear in the shrinker equation.The speed function enters indirectly when we rewrite the equation in graphical form; thespeed function adjusts for that the normal direction and vertical directions may not be thesame. The relative area element will be used to compute the mean curvature and to relatethe gradient of F to φ = h x, n i − H .A.1. Calculations.
The next lemma gives the expressions for the ν u , η u and w u on a graphΣ u over a general hypersurface Σ. The statement is rather technical and it is helpful to keepin mind the special case where Σ is the hyperplane R n and the quantities are given by ν u = p |∇ u | = w u and η u = u − h p, ∇ u i p |∇ u | . (A.2)The first part of the lemma gives similar formulas for a general Σ. The second part uses theformulas to compute Taylor expansions of the quantities. Some of these computations areused to compute linear approximations here, while others are not used in this paper but arerecorded for future reference and will be used elsewhere. Lemma A.3.
There are functions w, ν, η depending on ( p, s, y ) ∈ Σ × R × T p Σ that aresmooth for | s | less than the normal injectivity radius of Σ so that: w u ( p ) = w ( p, s, y ) = q | B − ( p, s )( y ) | , (A.4) ν u ( p ) = ν ( p, s, y ) = w ( p, s, y ) det ( B ( p, s )) , (A.5) η u ( p ) = η ( p, s, y ) = h p, n ( p ) i + s − h p, B − ( p, s )( y ) i w ( p, s, y ) , (A.6)where the linear operator B ( p, s ) ≡ Id − s A ( p ). Finally, the functions w , ν , and η satisfy: • w ( p, s, ≡ ∂ s w ( p, s,
0) = 0, ∂ y α w ( p, s,
0) = 0, and ∂ y α ∂ y β w ( p, ,
0) = δ αβ . • ν ( p, ,
0) = 1; the non-zero first and second order terms are ∂ s ν ( p, ,
0) = H ( p ), ∂ s ν ( p, ,
0) = H ( p ) − | A | ( p ), ∂ p j ∂ s ν ( p, ,
0) = H j ( p ), and ∂ y α ∂ y β ν ( p, ,
0) = δ αβ . • η ( p, ,
0) = h p, n i , ∂ s η ( p, ,
0) = 1, and ∂ y α η ( p, ,
0) = − p α . Proof.
Let ( p, s ) be Fermi coordinates on the normal tubular neighborhood of Σ, so that s measures the signed distance to Σ. If we fix an s and a path γ ( t ) in Σ, then applyingthe normal exponential map for time s sends γ ( t ) to γ ( t ) + s n ( γ ( t )). It follows that thedifferential is given by the symmetric linear operator(A.7) B ( p, s ) ≡ (Id − s A ( p )) : T p Σ → T p Σ , where we used that − A is the differential of the Gauss map to differentiate n and the Gausslemma to identify T p Σ with the tangent space to the level set of the distance to Σ.We will use this to compute the relative area element for the graph Σ u . Pushing forwardan orthonormal frame e i for Σ at p gives a frame E i for Σ u at ( p, u ( p ))(A.8) E i ≡ B ( p, u )( e i ) + u i ( p ) ∂ s . Thus, the metric on the graph is given in this frame by g uij ( p ) ≡ h E i , E j i = h B ( p, u )( e i ) , B ( p, u )( e j ) i + u i u j . (A.9)Since the e i ’s are orthonormal on Σ, we get ν u ( p ) = det (cid:0) B ( p, u ( p )) + ∇ u ⊗ ∇ u ( p ) (cid:1) . (A.10)Similarly, using the frame (A.8), we see that the vector field(A.11) ∂ s − B − ( p, u ( p ))( ∇ u )( p ) = e n +1 − B − ( p, u ( p ))( ∇ u )( p )is normal to Σ u . It follows that the speed function is given by w u ( p ) = h e n +1 , n u i − = | e n +1 − B − ( p, u ( p ))( ∇ u )( p ) |h e n +1 , e n +1 − B − ( p, u ( p ))( ∇ u )( p ) i = q | B − ( p, u ( p ))( ∇ u ( p )) | . (A.12)To rewrite the relative area element, we will need two elementary facts. The first is that for n × n matrices M and M , we have det( M M ) = det( M ) det( M ). The second is that fora vector v ∈ R n , we have(A.13) det(Id + v ⊗ v ) = 1 + | v | . Using these two facts, we now rewrite (A.10) as ν u ( p ) = det (cid:8) B ( p, u ( p )) (cid:0) Id + B − ( p, u ( p ))( ∇ u ( p )) ⊗ B − ( p, u ( p ))( ∇ u ( p )) (cid:1) B ( p, u ( p )) (cid:9) = [det ( B ( p, u ( p ))) w u ( p )] . (A.14)To compute the support function η u , first use the formula (A.11) to get(A.15) n u = e n +1 − B − ( p, u ( p ))( ∇ u )( p ) | e n +1 − B − ( p, u ( p ))( ∇ u )( p ) | = e n +1 − B − ( p, u ( p ))( ∇ u )( p ) w u ( p ) , where n u is evaluated at p + u ( p ) n ( p ). Thus, the support function is given by w u ( p ) η u ( p ) = h p + u ( p ) n ( p ) , e n +1 − B − ( p, u ( p ))( ∇ u )( p ) i (A.16) = h p, n ( p ) i + u ( p ) − h p, B − ( p, u ( p ))( ∇ u ( p )) i , where the last equality used that n ( p ) is equal to e n +1 at the point p + s n ( p ) for any s .We have now established the formulas (A.4), (A.5) and (A.6) for the functions w , ν , and η . It is clear from the expressions for w , ν and η that they are smooth in the three variablesprovided that s is sufficiently small.The next thing is to establish the second set of three claims that give the second orderTaylor expansions for w , ν , and η . The function w appears in all three expressions, so itis convenient to start there. It follows immediately that w ( p, s,
0) = 1. To compute thepartials involving y α ’s, we get(A.17) ∂ y α w ( p, s, y ) = P β ( B − ) αβ ( p, s ) y β w ( p, s, y ) . It follows that ∂ y α w ( p, s,
0) = 0. To get the Hessian, we differentiate (A.17) again(A.18) ∂ y α ∂ y β w ( p, ,
0) = ( B − ) αβ ( p, w ( p, ,
0) = δ αβ , where the last equality used that B ( p,
0) = Id.Using (A.5), we have ν ( p, s, y ) = w ( p, s, y ) B ( p, s ) where(A.19) B ( p, s ) = det ( B ( p, s )) = det (Id − s A ( p )) . We have B ( p, ≡ ∂ s B ( p,
0) = − Tr( A ( p )) = H ( p ). This also gives ∂ s ∂ p j B ( p,
0) = H j ( p ). To get the second derivative in s , observe that ∂ s log B ( p, s ) = Tr (cid:2) B − ( p, s ) ∂ s B ( p, s ) (cid:3) = − Tr (cid:2) (Id − s A ( p )) − A ( p ) (cid:3) . (A.20)Thus, we see that(A.21) ∂ s B ( p,
0) = ( ∂ s B ( p, H ( p ) − B ( p, | A | ( p ) = H ( p ) − | A | ( p ) . Combining the calculations for B with the earlier ones for w , we can compute the first threeTaylor series terms for ν . The constant term is ν ( p, ,
0) = 1. The first order terms are ∂ p j ν ( p, ,
0) = 0 , (A.22) ∂ s ν ( p, ,
0) = ( ∂ s B ( p, w ( p, ,
0) + ( ∂ s w ( p, , B ( p,
0) = H ( p ) , (A.23) ∂ y α ν ( p, ,
0) = ( ∂ y α w ( p, , B ( p,
0) = 0 . (A.24)The second order terms involving just s and p derivatives are simplified greatly since w ( p, s, ≡
1. These are ∂ p j ∂ p k ν ( p, ,
0) = 0 and ∂ s ν ( p, ,
0) = (cid:8)(cid:0) ∂ s B (cid:1) w + 2 ( ∂ s B ) ∂ s w + (cid:0) ∂ s w (cid:1) B (cid:9) ( p, ,
0) = ∂ s B ( p, H ( p ) − | A | ( p ) , (A.25) ∂ p j ∂ s ν ( p, ,
0) = (cid:8) ( ∂ s B ) ∂ p j w + (cid:0) ∂ p j ∂ s w (cid:1) B + (cid:0) ∂ p j ∂ s B (cid:1) w + ( ∂ s w ) ∂ p j B (cid:9) ( p, , ∂ p j ∂ s B ( p,
0) = H j ( p ) . (A.26) To compute the terms involving y derivatives, it is useful to keep in mind that B does notdepend on y . We get ∂ p j ∂ y α ν ( p, ,
0) = (cid:8)(cid:0) ∂ p j ∂ y α w (cid:1) B + ( ∂ y α w ) ∂ p j B (cid:9) ( p, ,
0) = 0 , (A.27) ∂ s ∂ y α ν ( p, ,
0) = { ( ∂ s ∂ y α w ) B + ( ∂ y α w ) ∂ s B} ( p, ,
0) = 0 , (A.28) ∂ y β ∂ y α ν ( p, ,
0) = (cid:0) ∂ y β ∂ y α w ( p, , (cid:1) B ( p,
0) = δ αβ . (A.29)Finally, using (A.6) and the fact that the first derivatives of w vanish at ( p, , η . (cid:3) A.2.
The mean curvature and its linearization via the first variation.
We will com-pute the mean curvature H u using the first variation of the area of Σ u . This gives a divergenceform equation in u . Corollary A.30.
The mean curvature H u of Σ u is given by H u ( p ) = wν [ ∂ s ν − div Σ ( ∂ y α ν )](A.31) = wν (cid:0) ∂ s ν − ∂ p α ∂ y α ν − ( ∂ s ∂ y α ν ) u α ( p ) − (cid:0) ∂ y β ∂ y α ν (cid:1) u αβ ( p ) (cid:1) , where w , ν and their derivatives are all evaluated at ( p, u ( p ) , ∇ u ( p )). Proof.
By Lemma A.3, the area of the graph Σ u is(A.32) Area(Σ u ) = Z Σ ν u dp Σ = Z Σ ν ( p, u ( p ) , ∇ u ( p )) dp Σ . Given a one-parameter family of graphs Σ u + tv with v compactly supported, differentiatingthe area gives ddt (cid:12)(cid:12) t =0 Area(Σ u + tv ) = Z Σ { ∂ s ν ( p, u ( p ) , ∇ u ( p )) v ( p ) + ∂ y α ν ( p, u ( p ) , ∇ u ( p )) v α ( p ) } dp Σ = Z Σ { ∂ s ν ( p, u ( p ) , ∇ u ( p )) − div Σ ( ∂ y α ν ( p, u ( p ) , ∇ u ( p ))) } v ( p ) dp Σ . (A.33)On the other hand, the variation vector field on Σ u is given by v e n +1 so the first variationformula (see, e.g., (1 .
45) in [CM3]) gives ddt (cid:12)(cid:12) t =0 Area(Σ u + tv ) = Z Σ u H u h v e n +1 , n u i = Z Σ H u ( p ) v ( p ) ν u ( p ) w u ( p ) dp Σ , (A.34)where the second equality used the definition of the speed function w u = h e n +1 , n u i − .Equating these two expressions for the derivative of area, we conclude that H u ( p ) ν ( p, u ( p ) , ∇ u ( p )) w ( p, u ( p ) , ∇ u ( p )) = ∂ s ν ( p, u ( p ) , ∇ u ( p )) − div Σ ( ∂ y α ν ( p, u, ∇ u )) . (A.35)This gives the first equality in (A.31); the second equality follows from the chain rule. (cid:3) A.3.
The F functional near a cylinder. We now specialize to where Σ is a cylinder in C k and F ( u ) is the F functional of the graph Σ u . Lemma A.36.
If Σ ∈ C k , then the gradient M ( u ) of the F functional is given by M ( u ) = νw (cid:18) H u − η (cid:19) e − √ k u + u , (A.37)where H u is the mean curvature of Σ u and ν, w, η are all evaluated at ( p, u ( p ) , ∇ u ( p )). Proof.
Since we are using the Gaussian L inner product, M ( u ) is defined by ddt (cid:12)(cid:12) t =0 F ( u + tv ) = (4 π ) − n Z Σ v M ( u ) e − | p | dµ Σ . (A.38)On the other hand, the first variation formula for the F functional from [CM1] gives ddt (cid:12)(cid:12) t =0 F ( u + tv ) = (4 π ) − n Z Σ u h v e n +1 , n u i (cid:18) H u − h n u , x i (cid:19) e − | x | dµ Σ u , (A.39)where each quantity is evaluated on Σ u . Given p ∈ Σ, we have | p + u ( p ) n ( p ) | = | p | + u + 2 u h p, n i = | p | + u + 2 √ k u , (A.40)where the last equality used that Σ ∈ C k . Writing (A.39) as an integral over Σ gives ddt (cid:12)(cid:12) t =0 F ( u + tv ) = (4 π ) − n Z Σ vw (cid:18) H u − η (cid:19) e − √ k u + u ν e − | p | dµ Σ . (A.41)The lemma follows by equating (A.38) and (A.41) (cid:3) Proof of Lemma 4.6.
By Lemma A.36 and Corollary A.30, M ( u ) can be written as M ( u ) e √ k u + u = ∂ s ν − ∂ p α ∂ y α ν − ( ∂ s ∂ y α ν ) u α ( p ) − (cid:0) ∂ y β ∂ y α ν (cid:1) u αβ ( p ) − ν w η . (A.42)Since the exponential term depends only on u , we have to show that each of the five termson the right side can be expressed as either:(i) f ( u, ∇ u ) , (ii) h p, V ( u, ∇ u ) i or (iii) Φ αβ ( u, ∇ u ) u αβ . The proof will repeatedly use the calculations from Lemma A.3.The key point is that A is parallel on cylinders and, thus, the linear operator B ( p, s )depends only on s (and not p ). In particular, the function ν depends only on s and y (andnot p ). Thus, the first three terms on the right side of (A.42) are type (i) and the fourthterm is type (iii). Similarly, w depends only on s and y , it suffices to show that w η is a sumof terms of the three allowed types. Lemma A.3 gives w η = h p, n ( p ) i + s − h p, B − ( p, s )( y ) i . (A.43)The first term is constant (so trivially type (i)) and the second is also type (i). Finally, since B depends only on s , the third term is type (ii). (cid:3) A.4.
Rescaled MCF near a shrinker.
Let Σ ⊂ R n +1 be an embedded shrinker and u ( p, t )a smooth function on Σ × ( − ǫ, ǫ ), giving a one-parameter family of hypersurfaces Σ u . Wenext derive the graphical rescaled MCF equation. Lemma A.44.
The graphs Σ u flow by rescaled MCF if and only if u satisfies ∂ t u ( p, t ) = w ( p, u ( p, t ) , ∇ u ( p, t )) (cid:18) η ( p, u ( p, t ) , ∇ u ( p, t )) − H u (cid:19) . (A.45) Proof.
As in [EH], the rescaled MCF equation x t = (cid:0) h x, n i − H (cid:1) n is equivalent (up totangential diffeomorphisms) to the equation(A.46) ( x t ) ⊥ = 12 h x, n i − H .
The variation vector field and unit normal for Σ u are ∂ t u ( p, t ) n ( p ) and n u , respectively, atthe point p + u ( p, t ) n ( p ), so we get the equation h n ( p ) , n u i ∂ t u ( p, t ) = h ( ∂ t u ( p, t )) n ( p ) , n u i = 12 η u − H u . (A.47)Finally, multiplying through by w u = h n ( p ) , n u i − gives the lemma. (cid:3) We will use the following lemma bounding the distance between time slices of a rescaledMCF by the L norm of the gradient of the F functional. Lemma A.48.
Given n , there exist C and δ > ∈ C k and Σ u is a graphicalsolution of rescaled MCF on [ t , t ] with k u ( · , t ) k C ≤ δ , then Z Σ | u ( p, t ) − u ( p, t ) | e − | p | ≤ C Z t t Z Σ u ( t = r ) (cid:12)(cid:12)(cid:12)(cid:12) h x, n i − H (cid:12)(cid:12)(cid:12)(cid:12) e − | x | dr . (A.49) Proof.
By Lemma A.44, u satisfies ∂ t u ( p, t ) = w ( p, u ( p, t ) , ∇ u ( p, t )) (cid:18) η ( p, u ( p, t ) , ∇ u ( p, t )) − H u (cid:19) . (A.50)Since | u | and |∇ u | are small, Lemma A.3 gives that both w and the relative area element ν u are uniformly bounded and (A.40) relates the Gaussians on Σ and Σ u , so we get Z Σ | ∂ t u ( p, t ) | e − | p | ≤ C Z Σ (cid:12)(cid:12)(cid:12)(cid:12) η ( p, u ( p, t ) , ∇ u ( p, t )) − H u (cid:12)(cid:12)(cid:12)(cid:12) e − | p | ≤ C ′ Z Σ (cid:12)(cid:12)(cid:12)(cid:12) η ( p, u ( p, t ) , ∇ u ( p, t )) − H u (cid:12)(cid:12)(cid:12)(cid:12) ν u e − | p + u ( p,t ) n | (A.51) = C ′ Z Σ u (cid:12)(cid:12)(cid:12)(cid:12) h x, n i − H (cid:12)(cid:12)(cid:12)(cid:12) e − | x | . The lemma follows from integrating this with respect to t , using the fundamental theoremof calculus and Fubini’s theorem. (cid:3) Appendix B. An interpolation inequality
We will use the following interpolation inequality which is well-known, but we are includingthe short proof since we do not have an exact reference. Unlike the rest of this paper, the L norms below are unweighted. Lemma B.1.
There exists C = C ( k, n ) so that if u is a C k function on B r ⊂ R n , then k u k L ∞ ( B r ) ≤ C n r − n k u k L ( B r ) + k u k a k,n L ( B r ) k∇ k u k − a k,n L ∞ ( B r ) o , (B.2) r k∇ u k L ∞ ( B r ) ≤ C n r − n k u k L ( B r ) + r k u k b k,n L ( B r ) k∇ k u k − b k,n L ∞ ( B r ) o , (B.3) r k∇ u k L ∞ ( B r ) ≤ C n r − n k u k L ( B r ) + r k u k c k,n L ( B r ) k∇ k u k − c k,n L ∞ ( B r ) o , (B.4)where a k,n = kk + n , b k,n = k − k + n and c k,n = k − k + n . Proof.
By scaling, it suffices to prove the case r = 1.The starting point is the following standard consequence of the Bernstein/Kellogg inequal-ity for polynomials, [K]:(K) Given n and d , there exists C d,n so that if p is a polynomial of degree at most d ona ball B δ ⊂ R n for some δ >
0, then k p k L ∞ ( B δ ) + δ k∇ p k L ∞ ( B δ ) + δ k∇ p k L ∞ ( B δ ) ≤ C d,n δ − n Z B δ | p | . (B.5)Set m = k∇ k u k L ∞ ( B ) . Choose x ∈ B where | u | achieves its maximum and let p be thedegree ( k −
1) polynomial giving the first ( k −
1) terms of the Taylor series of u at x . Inparticular, given any δ ∈ (0 , Z B δ ( x ) | u − p | ≤ C m δ n + k , (B.6)where C depends on n and k . Using this in (K) gives k u k L ∞ ( B ) = | p | ( x ) ≤ C δ − n Z B δ ( x ) | p | ≤ C δ − n (cid:26)Z B δ ( x ) | u | + Z B δ ( x ) | u − p | (cid:27) ≤ C δ − n (cid:8) k u k L ( B ) + C m δ n + k (cid:9) . (B.7)We now consider two cases. First, if m ≤ k u k L ( B ) , then (B.7) with δ = 1 gives k u k L ∞ ( B ) ≤ C k u k L ( B ) . (B.8)Next, if m > k u k L ( B ) , then we set δ n + k = k u k L B m (which is less than one) and (B.7) gives k u k L ∞ ( B ) ≤ C k u k kn + k L ( B ) m nn + k . (B.9)Thus, we see that (B.2) holds in either case.We will argue similarly to get the ∇ u bound. This time, let x ∈ B be a point where |∇ u | achieves its maximum. Given δ ∈ (0 , |∇ u | ( x ) = |∇ p | ( x ) ≤ C δ − n − (cid:8) k u k L ( B ) + C m δ n + k (cid:9) . (B.10) In the case where m ≤ k u k L ( B ) , we get (B.3) by setting δ = 1. On the other hand, when m > k u k L ( B ) , then we set δ n + k = k u k L B m (which is less than one) and (B.10) gives |∇ u | ( x ) ≤ C k u k k − n + k L ( B ) m n +1 n + k , (B.11)completing the proof of (B.3). The last bound (B.4) follows similarly. (cid:3) References [Al] W.K Allard,
On the first variation of a varifold . Ann. of Math. (2) 95 (1972), 417–491.[AA] W.K. Allard and F.J. Almgren, Jr,
On the radial behavior of minimal surfaces and the uniqueness oftheir tangent cones . Ann. of Math. (2) 113 (1981), no. 2, 215–265.[AAG] S. Altschuler, S.B. Angenent, and Y. Giga,
Mean curvature flow through singularities for surfaces ofrotation , Journal of geometric analysis, Volume 5, Number 3, (1995) 293–358.[An] B. Andrews,
Noncollapsing in mean-convex mean curvature flow . Geom. Topol. 16 (2012), no. 3, 1413–1418.[Be] P.A. Beck, Metal Interfaces, p. 208. Cleveland, Ohio: American Society for Testing Materials, 1952.[B] K. Brakke,
The motion of a surface by its mean curvature . Mathematical Notes, 20. Princeton UniversityPress, Princeton, N.J., 1978.[BE] D. Bakry and M. ´Emery,
Diffusions hypercontractives . S´eminaire de probabilit´es, XIX, 1983/84, 177–206, Lecture Notes in Math., 1123, Springer, Berlin, 1985.[Br] S. Brendle,
An inscribed radius estimate for mean curvature flow in Riemannian manifolds ,http://arXiv:1310.3439.[BrCoL] H. Brezis, J.-M. Coron and E. Lieb,
Harmonic maps with defects , Comm. Math. Phys. 107 (1986),no. 4, 649–705.[Bu] J. Burke,
Some factors affecting the rate of grain growth in metals , AIME Transactions, (1949) vol.180, pp. 73–91.[CIM] T.H. Colding, T. Ilmanen and W.P. Minicozzi II,
Rigidity of generic singularities of mean curvatureflow , preprint, arXiv:1304.6356.[CIMW] T.H. Colding, T. Ilmanen, W.P. Minicozzi II and B. White,
The round sphere minimizes entropyamong closed self-shrinkers , J. Differential Geom., vol. 95, no. 1 (2013) 53–69.[CM1] T.H. Colding and W.P. Minicozzi II,
Generic mean curvature flow I; generic singularities , Annals ofMath., Volume 175 (2012), Issue 2, 755–833.[CM2] ,
Smooth compactness of self-shrinkers , Comm. Math. Helv., 87 (2012) 463–475.[CM3] ,
A course in minimal surfaces , Graduate Studies in Mathematics, Vol. 121, AMS (2011).[CM4] ,
On uniqueness of tangent cones for Einstein manifolds , Invent. Math., to appear,http://arXiv:1206.4929.[CM5] ,
The singular set of mean curvature flow with generic singularities , preprint.[CM6] ,
Lojasiewicz inequalities and applications , preprint, http://arXiv:1402.5087.[CMP] T.H. Colding, W.P. Minicozzi II and E.K. Pedersen,
Mean curvature flow as a tool to study topologyof 4-manifolds , preprint, http://arXiv:1208.5988.[EH] K. Ecker and G. Huisken,
Interior estimates for hypersurfaces moving by mean curvature . Invent. Math.105 (1991), no. 3, 547–569.[FFl] H. Federer and W. Fleming,
Normal and integral currents , Ann. of Math. 72 (1960), 458–520.[GK] Z. Gang and D. Knopf,
Universality in mean curvature flow neckpinches ,http://arxiv.org/abs/1308.5600.[GKS] Z. Gang, D. Knopf, and I.M. Sigal,
Neckpinch dynamics for asymmetric surfaces evolving by meancurvature flow , preprint, http://arXiv:1109.0939v1.[GS] Z. Gang and I.M. Sigal,
Neck pinching dynamics under mean curvature flow , preprint,http://arxiv.org/abs/0708.2938. [GGS] M. Giga, Y. Giga, and J. Saal,
Nonlinear partial differential equations. Asymptotic behavior of solu-tions and self-similar solutions . Progress in Nonlinear Differential Equations and their Applications, 79.Birkh¨auser Boston, Inc., Boston, MA, 2010.[Gr] A. Grigor’yan,
Heat kernel and analysis on manifolds . AMS/IP Studies in Advanced Mathematics, 47.American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.[Hr] R.M. Hardt,
Singularities of harmonic maps . Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 1, 15–34.[HrLi] R.M. Hardt and F.-H. Lin,
Stability of singularities of minimizing harmonic maps. J. DifferentialGeom . 29 (1989), no. 1, 113–123.[HaP] D. Harker and E. Parker,
Grain shape and grain growth . Trans. Am. Soc. Met. (1945) 34:156–201.[HaK] R. Haslhofer and B. Kleiner,
Mean curvature flow of mean convex hypersurfaces , preprint,http://arXiv:1304.0926.[H1] G. Huisken,
Asymptotic behavior for singularities of the mean curvature flow . J. Differential Geom. 31(1990), no. 1, 285–299.[H2] ,
Local and global behaviour of hypersurfaces moving by mean curvature . Differential geometry:partial differential equations on manifolds (Los Angeles, CA, 1990), 175–191, Proc. Sympos. Pure Math.,54, Part 1, Amer. Math. Soc., Providence, RI, 1993.[H3] ,
Flow by mean curvature of convex surfaces into spheres , J. Differential Geometry 20 (1984)237-266.[HP] G. Huisken and A. Polden,
Geometric evolution equations for hypersurfaces . Calculus of variations andgeometric evolution problems, 45–84, Lecture Notes in Math., 1713, Springer, Berlin, 1999.[HS1] G. Huisken and C. Sinestrari,
Convexity estimates for mean curvature flow and singularities of meanconvex surfaces , Acta Math. 183 (1999) no. 1, 45–70.[HS2] G. Huisken and C. Sinestrari,
Mean curvature flow singularities for mean convex surfaces , Calc. Var.Partial Differ. Equ. 8 (1999), 1–14.[I1] T. Ilmanen,
Singularities of Mean Curvature Flow of Surfaces
Elliptic regularization and partial regularity for motion by mean curvature , Memoirs Amer.Math. Soc. 520 (1994).[K] O. Kellogg,
On bounded polynomials in several variables , Math. Z. 27 (1928), no. 1, 55–64.[L] S. Lojasiewicz,
Ensembles semi-analytiques , IHES notes (1965).[M] W.W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27, 900–904 (1956).[N] J. von Neumann, in Metal Interfaces (ed. Herring, C.) 108110 (American Society for Metals, Cleveland,1952).[Sc] F. Schulze,
Uniqueness of compact tangent flows in mean curvature flow , Crelle, to appear,arXiv:1107.4643.[Se] N. Sesum,
Rate of convergence of the mean curvature flow . Comm. Pure Appl. Math. 61 (2008), no. 4,464–485.[Si1] L. Simon,
Asymptotics for a class of evolution equations, with applications to geometric problems ,Annals of Math. 118 (1983), 525–571.[Si2] ,
A general asymptotic decay lemma for elliptic problems . Handbook of geometric analysis. No.1, 381–411, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008.[Si3] ,
Rectifiability of the singular set of energy minimizing maps , Calc. Var. Partial. DifferentialEquations 3 (1995), no. 1, 1–65.[Si4] ,
Theorems on regularity and singularity of energy minimizing maps , Lectures on geometricvariational problems (Sendai, 1993), 115–150, Springer, Tokyo, 1996.[Si5] ,
Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizingmaps , Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 246–305, Int. Press, Cambridge,MA, 1995.[SS] H. Soner and P. Souganidis,
Singularities and uniqueness of cylindrically symmetric surfaces moving bymean curvature . Comm. Partial Differential Equations 18 (1993), no. 5-6, 859–894.[Su] T. Sutoki,
On the mechanism of crystal growth by annealing , Scientific Reports of Tohoku. ImperialUniversity, (1928) vol. 17, no. 2, pp. 857–876. [T] J. Taylor, Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 in R ,Invent. Math. 22 (1973), 119–159.[W1] B. White, The nature of singularities in mean curvature flow of mean-convex sets . J. Amer. Math.Soc. 16 (2003), no. 1, 123–138.[W2] ,
Evolution of curves and surfaces by mean curvature . Proceedings of the International Congressof Mathematicians, Vol. I (Beijing, 2002), 525–538.[W3] ,
A local regularity theorem for mean curvature flow . Ann. of Math. 161 (2005), 1487–1519.[W4] ,
The mathematics of F. J. Almgren, Jr.
J. Geom. Analysis 8 (1998), no. 5, 681–702.[W5] ,
Partial regularity of mean-convex hypersurfaces flowing by mean curvature , Int. Math. Res.Notices (1994) 185–192.
MIT, Dept. of Math., 77 Massachusetts Avenue, Cambridge, MA 02139-4307.
E-mail address ::