T. H. COLDING AND W. P. MINICOZZI S UNIQUENESS OF BLOWUPS AND LOJASIEWICZ INEQUALITIES

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1 T. H. COLDING AND W. P. MINICOZZI S UNIQUENESS OF BLOWUPS AND LOJASIEWICZ INEQUALITIES CHRISTOS MANTOULIDIS Abstract. In March 01 I gave a thorough lecture series at Stanford on Toby Colding and Bill Minicozzi s 013 paper titled Uniqueness of Blowups and Lojasiewicz Inequalities. These notes served as a guideline for my presentation. I was asked if I could upload them, so here they are. They go through the entire paper in depth, and contain proofs that were as licit as I could make them be for my presentation. There are some minor ository differences between these notes and the original paper. These changes were made solely for my own convenience and better understanding. I do not claim originality over the contents of the proofs, except perhaps for any mistakes I may have accidentally introduced here and there. I would like to take this opportunity to thank Toby and Bill for having been very responsive and very kind in our correspondence over the past few monthts. Contents 1. Introduction Mean Curvature Flow Sign and notation conventions F functional, entropy Parabolic rescaling 1.5. Monotonicity 1.6. Mean convex flows 1.7. Cylindrical flows are generic 1.8. Question of uniqueness 1.9. Rescaled Mean Curvature Flow. Synopsis 3.1. Ultimate goal 3.. Graphical and Entropy scales 3.3. Lojasiewicz inequalities.. Everything put together 5.5. A note on F and its variations 6 3. First Lojasiewicz Inequality LI1 on spheres Dimension reduction Vertical translation Cylindrical estimates Proof of First Lojasiewicz inequality 10. Second Lojasiewicz Inequality Understanding the kernel of L 1.. Preliminary gradient estimate 1.3. Proof of Second Lojasiewicz inequality Scale Comparison Theorem Improvement argument Extension argument Proof of Scale comparison theorem Date: June 1, 01. 1

2 1. Introduction 1.1. Mean Curvature Flow. MCF is the evolution equation τ x = H for closed manifolds M n τ R n+1. We assume that the first singular time is τ = 0, and thus that τ < 0 everywhere below. We use τ for the time parameter because we will predominantly work with a modified flow, for which we reserve the t parameter. 1.. Sign and notation conventions. 1 n is the outer unit normal. H is the mean curvature scalar H > 0 on spheres, but also possibly the mean curvature vector depending on the context F functional, entropy. For M n R n+1 we define F M = π n/ x dv x M Occasionally we write F x0,λm for F λ 1 M x 0. Under a normal perturbation M M + ψn, the first variation of F is: F M ψ = π n/ ψ H 1 x n x dv x M The quantity 1 x n H is denoted by φ. Notice that this is the negative of the Euler-Lagrange operator with respect to the Gaussian measure dµx = e x / dv x. We define the entropy of M n to be λm sup x0,λ F x0,λm. Bounded entropy is important because it gives a Hölder inequality on the Gaussian weighted L p spaces and also that F M \ B R Cn, γ λ 0 R ρn γ R for any γ 0, 1. This allows us to localize our estimates to compact sets. 1.. Parabolic rescaling. Parabolic rescaling is best phrased in terms of spacetime flows. Let M be a spacetime MCF for t < 0, and let λ > 0. Then λ # M {λ y, λ t : y, t M} is another MCF, said to be a parabolic dilation at 0, 0. If λ i and 0, 0 M, then after possibly passing to a subsequence M limλ i # M i is an ancient Brakke flow which is in fact backwards self-similar, i.e. M λ = λ M 1 for any λ > 0. The limit M, called a tangent flow, is likely to depend on the particular choice of λ i Monotonicity. Huisken s monotonicity can be ressed as follows: the parabolic density function ΘM, x 0, t 0, λ πλ M n/ x x 0 t0 λ λ dv x is increasing in λ > 0. Notice that ΘM, x 0, t 0, λ = F λ 1 M t0 λ x 0. We also write ΘM, X lim λ 0 ΘM, X, λ Mean convex flows. A flow M is called mean convex if the initial time slice satisfies H 0. Brian White has shown that all tangent flows at the first singular time of a mean convex flow are elements of C k { all rotations of round cylinders S k k R n k R n+1 } i.e., all tangent flows of a mean convex flow are cylindrical.

3 1.7. Cylindrical flows are generic. Colding, Ilmanen, and Minicozzi have shown that cylindrical flows are generic in some sense. In particular, if one tangent flow at a point is cylindrical, then they all are. Furthermore, there exists a neighborhood of the point in which any other singularities are also cylindrical Question of uniqueness. There remains the question of uniqueness for cylindrical flows: do we always get the same R n k axis independently of how we dilate the flow? In this paper, Colding and Minicozzi answer this in the affirmative Rescaled Mean Curvature Flow. Colding and Minicozzi work in the context of RMCF. RMCF is the evolution equation t x = H + 1 x = φn for closed manifolds Σ t. RMCF is the negative L -gradient flow of F, i.e. along dµ Σ F = φ. Furthermore t λσ t decreases along a RMCF. The stationary points of RMCF are called shrinkers and are those for which φ = 0 H = 1 x n. RMCF is obtained from MCF by setting t = log τ, Σ t = 1 τ M τ and in particular exists even for t. Notice that picking t i in the RMCF corresponds to studying the time t = 1 slice of limit tangent flow. As such, the sequence t Σ t may have many limit points. In other words, there is only one tangent flow if and only if t Σ t has a unique limit. This is what we re going to show.. Synopsis.1. Ultimate goal. Uniqueness will follow from the following package theorem. Theorem 6.1. There exist K > 0 large, R > 0 large, ε 0 > 0 small, and τ > 0 small depending on n, λ 0 such that if 1 Σ s, s t 1, t + 1] is a RMCF, λσ s λ 0 for s t 1, t + 1], and 3 there is a fixed cylinder C C k over which Σ s B R is a graph with,α ε for all s t 1, t + 1] Then F Σ t F C k τ K F Σ t 1 F Σ t+1. Remark. The finite dimensional Lojasiewicz approach concerns the evolution equation ẋ = fx, where f is real analytic. One can prove that the curve t xt has finite total length, i.e. 0 ẋt dt < provided it has at least one limit point. Therefore that limit point is necessarily the unique limit point, i.e. t xt converges. The motivation is to extend this to infinitely many dimensions. Remark. The observation that s necessary to go from 6.1 to proving uniqueness is that for any ε 0 > 0 there exists δ > 0 such that if Σ t1 is C,α -ε 0 / close to a cylinder on some ball of arbitrary radius, and t t 1 φ] L 1 Σ t dt δ then every Σ t, t a, b], is a graph over the same cylinder inside the ball of the same radius. This is because we can control the L 1 distance between the graphs in terms of the integral of Σ F dt, since the evolution is in terms of Σ F. The L 1 bound improves to a Hölder estimate by parabolic Schauder theory. Once we have a result like this, then 0 φ] L 1 Σ t dt < will imply that for large times the surfaces are graphical over a unique cylinder. 3

4 Proof of: 6.1 Uniqueness. By a contradiction argument and White s theorem we know that for t large enough, the Σ s can all be written as a uniformly small cylindrical graph over a cylinder depending s t 1,t+1] only on t. Then by 6.1, F Σ t F C k τ K F Σ t 1 F Σ t+1 for all t large enough, and τ > 0 small and fixed. By a functional inequality Lemma 6. that translates this estimate into a polynomial decay as in the standard Lojasiewicz inequalities, F Σ t F C k C t 1/1 τ, we get 1/ F Σ j F Σ j+1 ] < At this point we can estimate 1 Σt F ] L 1 dt = 1 j=1 φ] L 1 Σ t dt F Σ 0 1/ j=1 F Σ j F Σ j+1 ] 1/ < and therefore the total length of the flow is finite, so there is a unique limit point... Graphical and Entropy scales. Definition Graphical radius. Let Σ n R n+1 be a given surface that is approximately cylindrical. Its graphical scale is the maximal radius up to which it is C,α close to a cylinder. For given ε 0 > 0 small, l N large, C l > 0 large, we define R ε0,l,c l graph Σ to be the largest radius R for which: 1 l A C l on Σ B R, and Σ B R is C,α ε 0 -close to a cylinder in C k. Remark. The constant ε 0 will be chosen universally by two lemmas that come from pure functional analysis and will depend only on n. It is l, C l that we have to be careful with. Definition Entropy radius. Let Σ s be a RMCF for s t 1, t + 1]. The entropy radius is an artificial radius that is large when we are close to a stationary point of F, and therefore to a cylinder Colding, Ilmanen, Minicozzi. It is defined indirectly at time t via the relation R entrσ t = F Σ t 1 F Σ t+1 Remark. The graphical scale is more convenient to work with analytically while the entropy radius is more natural to the problem. The key heuristic to have in mind is that: R entrσ t = F Σ t 1 F Σ t+1 δf t Σt F ] L φ] L Σ t R entrσ t φ] L We often run into ressions of the form: or powers thereof. Note the following points: φ] L Σ t B R + R 1 The goal is to write everything in terms of φ. We want to have φ] L B R be the dominant term; the non-compact error term should be swallowed into the φ]. Since φ] L Σ t R entrσ t /, we have a problem when R R entr Σ t : the noncompact error term dominates. 3 This is where the main scale comparison theorem Theorem 5.3 comes in. It says that the graphical radius subsumes the entropy radius. Theorem 5.3. There exist µ > 0 small, l 0 1 large, C > 0 large depending on n, λ 0, ε 0, such that if: 1 Σ t is a RMCF on 1, 1], and l l 0,

5 then there exists C l > 0 large depending on n, λ 0, ε 0, l, such that 1 + µr entr Σ 0 min 1/ t 1 Rε0,l,C l graph Σ t + C.3. Lojasiewicz inequalities. We will reach Theorem 6.1 through two Lojasiecz inequalities. The first one is based on very general cylindrical estimates that have nothing to do with MCF/RMCF. Theorem 0. - LI1. There exist ρ > 0 ρ = 5n + 10 and l 0 1 large depending on n, and a sequence c l,n 0, 1 1 as l, such that if: 1 Σ n R n+1 has λσ λ 0, l l 0, C l <, and 3 R < R ε0,l,c l graph, then inf w C } k] L C C Σ B R C Rρ{ φ] c l,n L 1 Σ B R + c l,n R k for C = Cn, λ 0, ε 0, l, C l, where w C denotes the distance function to the axis of the cylinder C. Remark. One should interpret LI1 as saying that the squared L closeness to an optimal cylinder can be controlled up to a power slightly weaker than one by the L 1 norm of φ alongside an onentially decaying noncompact error term. The second Lojasiewicz inequality requires more effort, but it will follow from the first Lojasiewicz inequality after careful analysis of the kernel K of the linearization L of Σ F. Theorem LI. There exists ρ > 0 and l 0 1 large depending on n, and a sequence c l,n 0, 1 1 as l such that if: then 1 Σ n R n+1 has λσ λ 0, l l 0, C l <, 3 R < R ε0,l,c l graph, and β 0, 1, { F Σ F C k C R ρ φ] c 3+β l,n 1+β L Σ B R + c l,n for C = Cn, λ 0, ε 0, l, C l. 3 + β 1 + β R β } R Remark. This corresponds to the gradient Lojasiewicz inequality. It will be crucial in the scale comparison theorem 5 in the paper as well as in the proof of the package theorem, 6.1. The first two terms should be understood as coming from LI1 when we are close to ker L i.e. the normal component to the kernel is reasonably small, and the last term is really a crude estimate that we can use when we are very far from the kernel of L i.e. the kernel component is much, much smaller than the normal component... Everything put together. In this subsection we show how LI and the scale comparison theorem imply the main theorem Theorem 6.1. There exist K > 0 large, R > 0 large, ε 0 > 0 small, and τ > 0 small depending on n, λ 0 such that if 1 Σ s, s t 1, t + 1] is a RMCF, λσ s λ 0 for s t 1, t + 1], and 3 there is a fixed cylinder C C k over which Σ s B R is a graph with,α ε for all s t 1, t + 1] Then F Σ t F C k τ K F Σ t 1 F Σ t+1. Proof of: By LI it is true that { F Σ t F C k C R ρ φ] c 3+β l,n 1+β L Σ s B R + c l,n β 1 + β R β } R

6 for the maximal choice of radius, R R ε0,l,c l. It suffices to bound each term on the right by a power greater than 1/ of δf t F Σ t 1 F Σ t+1. By the scale comparison theorem, graph R 1 + µ R entr Σ t C as long as C l > is large enough depending on l. The point is that µ, C are independent of l, since we ll pick l large in a little bit. Then the last term can be bounded by 3 + β C 3 + β δf t 3+β 1+µ R 1 + µ RentrΣ t Picking β 0, 1 close enough to 1 makes the onent larger than 1/. The first two terms on the RHS can be dealt with simultaneously, provided one observes a mean-value-type inequality saying that At this point we choose l large enough such that and the result follows. φ] L Σ t B R C F Σ t 1 F Σ t+1 = C δf t c l,n 3 + β 1 + β and c l,n1 + µ > 1.5. A note on F and its variations. We ve already shown that the Euler-Lagrange functional of F with respect to Gaussian measure, dµ, is dµ Σ F = φ = H 1 x n. If Σ n R n+1 is some hypersurface and u : Σ n R is small then the graphical surface Σ u has its own Euler-Lagrange functional of F with respect to Gaussian measure, φ u. If we like, we can pull that back to Σ. It will be convenient to pull back the Euler-Lagrange operator of F with respect to dv x instead. We denote that by Mu. When Σ is a cylinder, Mu = ν u H u 1 w u η u where 1 ν u is the relative volume element, H u is the mean curvature scalar of the cylinder, 3 η u is the support function, + u n, n u, and w u is the speed function dist Σ, n u 1. The linearization of Mu at u = 0 is the L operator, k u + u L = L + A + 1 = 1 x t + A When Σ is a cylinder, L = L + 1 = 1 x t First Lojasiewicz Inequality The first Lojasiewicz inequality states that φ correctly captures the L -closeness of Σ to a cylinder, up to a noncompact error term with onential decay. The plan of action to prove this is: 1 Show it first for the special case of almost spherical shrinkers. Those are compact. We get a stronger C,α inequality, and no error term. This is Lemma.5. Present a dimension reduction technique that can let us reduce the case of noncompact cylinders to that of compact spheres at the waist of the cylinders. This is Lemma Apply the strong C,α spherical inequality on the waist of the cylinder and then extend in the flat directions under the assumption that we indeed have reasonably flat directions. This gives a weaker C 1 inequality, and it is Proposition.1. 6

7 Formulate a criterion for the non-spherical directions to be sufficiently flat in terms of just φ] L 1. This is Corollary Put it all together, obtaining LI1 - Theorem LI1 on spheres. Lemma.5. There exist ε 0 > 0 small and C > 0 large depending on k, α 0, 1 such that if: 1 Σ k R k+1 is a graph over S k k with u ε 0, then u,α C φ 0,α. Proof sketch. The first three eigenvalues of L = + 1 on S k k are 1, 1, 1 k, so 0 is not an eigenvalue, so by Schauder theory u,α Ck, α Lu 0,α On the other hand L is the linearization of Σ F = φ, so φ Lu 0,α Ck, α u u,α provided we require u ε 0 to be uniformly bounded so the constant C is finite, and independent of ε 0. Therefore: φ Lu 0,α Ck, α ε 0 u,α By taking ε 0 0 depending on k, α we can use the second inequality inside the first, and the result follows. 3.. Dimension reduction. To reduce to the case of spheres we need a dimension reduction argument. The dimension reduction argument relies on the idea that if we slice an almost-cylindrical almost-shrinker almost-orthogonally by a plane, then we get an almost-cylindrical almost-shrinker of one flat dimension lower. Lemma.11. Let Σ k R k+1, Σ 0 = Σ {x k+1 = 0}, x 0 Σ 0 be a transverse intersection point, and suppose ε 0, 1/. If 1 t x k+1 1 ε at x 0, t t x k+1 ε at x 0, and 3 the tensor norms A, t x k+1 + A, t x k+1 ε on the indicated restricted almost-flat directions at x 0, then if φ 0 is the φ-function for Σ 0, for a universal constant C > 0. φ φ 0 + Σ0 φ φ 0 C ε 1 + φ + φ Sketch of proof. The proof is reasonably straightforward. We compute φ φ 0 = v 1 φ + 1 v v v A v, v v where v = t x k+1 near x 0. Immediately we see φ φ 0 C ε + C ε φ. By differentiating along Σ 0 and using Kato s inequality we get Σ0 φ φ 0 C ε + C ε φ + C ε φ Vertical translation. The point now is to apply the spherical inequality on the waist of a cylinder and the translate all bounds vertically. Proposition.1. There exist ε 1, ε > 0 small and C > 0 large depending on n, δ > 0, M > 0, so that if 1 H δ > 0, A + A M on Σ B R, Σ B 5 n is C ε 1 -close to a cylinder in C k, k 1, and 3 5 n r < R is such that r φ 1,B5 n + r 5 A/H 1,Br ε then Σ B r 3k is a a graph over a cylinder in C k with } 1 C {r φ 1,B5 n + r 5 A/H 1,Br 7

8 Remark. 1 The first assumption is a crude assumption on a large scale. The second assumption is a small-scale fit that allows us to dimension-reduce to something reasonable. 3 The third assumption is twofold: the first term will get the waist to be C,α -close to S k k and the second term says that A/H is almost parallel and allows us to extend estimates in the flat directions. Remark. The radius 5 n is not arbitrary. We ect the waist to have radius k, so we re allowing for k n and simultaneously giving some extra wiggle room. Remark. It s actually better to state a stronger result, seeing as to how it s what is actually proved and is what is going to be invoked later. For y Σ B r 3k and u the graph function over the cylinder, } u y + u y C {r φ 1,B5 n + r 5 A/H 1,B y Sketch of proof of.1. We write τ for A/H, ε τ r = τ 1,Br, and ε φ r = φ 1,Br. Step 1. Setup. Pick p Σ B n arbitrary and fixed for the duration of the proof. Since Σ B 5 n is C ε 1 -close to a cylinder, by possibly shrinking ε 1 depending on n alone we can get an orthonormal basis of eigenvectors flat: v 1 p,..., v n k p, spherical: z 1 p,..., z k p with corresponding eigenvalues λ 1,..., λ n k, σ 1,..., σ k such that λ i 1 100n and σ j 1 n Let s also define np to be the outer unit normal at p. By a general lemma of Colding and Minicozzi from a different paper, having λ i and σ j far apart forces a better yet bound on the flat eigenvalues λ i in terms of τ 1 : λ i Cn, γ ε τ 5 n where the δ came in because we have to multiply through by H δ to get from A to τ = A/H. Define the linear coordinate functions f i x x v i p, g j x x z j p, g k+1 x x np, i n k, and j k and the tangent vectors v i x t f i x, i n k We will be slicing with the plane P = {f 1 =... = f n k = 0} and we will be interested in the cylinder normal to this plane. The radial distance to the axis of that cylinder is given by: k+1 wx g j x j=1 Without loss of generality we may translate our surface so that p P. Step. Bounds near p. For r > 5 n let Ω r = Σ B Σ 3rp, the latter ball being a geodesic ball. The v i enjoy the following properties in Ω r : 1 v i v i p Cn, δ, M r ε τ r, τv i Cn, δ, M r ε τ r, and 3 vi A Cn, δ, M r ε τ r. Let γt be a curve of length 3r starting at p, and let wt be the unit parallel translation of v i p along γt. Note that w = 1, γ 1, τ ε τ r on Ω r γt τwt ε τ r so so τwt length γ ε τ r + τv i p 3 r ε τ r + Cn, δ ε τ 5 n Cn, δ r ε τ r Since Rn+1 γ w = A γ, w n, the FTC gives Awt = H τwt Cn, δ, M r ε τ r. wt v i p length γ Cn, δ, M r ε τ r Cn, δ r ε τ r 8

9 Of course wt is some tangent vector x and v i x is the one closest to v i p, so the inequality above also holds true for v i v i p : Furthermore by the bound on τ, v i v i p length γ Cn, δ, M r ε τ r Cn, δ r ε τ r 1 τv i τv i τv i p + τv i p Cn, δ, M r ε τ r Equation 3 follows from by the Codazzi equation. Step 3. Dimension reduction. We seek to slice Σ with P = {f 1 =... = f n k = 0}. We use a modification of.11 that allows multiple slices at once. The point is that slicing by P gives a compact topological sphere that is C close to S k k, whereas slicing with lower dimensional things gives noncompact objects. The proof is the same for multiple simultaneous slices. The coordinates we are slicing with respect to are f 1,..., f n k so Σ 0 = Σ P is a compact topological sphere contained well inside B 5 n. We claim that the assumptions of.11 are satisfied on Σ 0 S k k Ω 5 n. Indeed, at any point x on the topological sphere: 1 t f i t f i p t f i t f i p 1 C r 0 ε τ r 0, for X T x Σ, X t f i = X f i f i nn = X f i n n f i n X n t X t f i = f i n AX ] = f i t f i n AX so t t f i C r 0 ε τ r 0 ; finally, 3 A, t f i + A, t f i C r 0 ε τ r 0. We re writing r 0 = 5 n above. In view of the lower bound r 5 n, as long as we require ε to be small enough depending on n, δ, M to make the constants above comparable to 1/, we can apply.11 and get that Σ 0 is C,α ε φ + ε τ -close to S k k. Therefore the radial axis distance function satisfies: w k,α,σ0 Cn, δ, M ε φ 5 n + ε τ 5 n Step. Extending vertically. We extend the bound by constructing a vertical radial flow with speed Notice that f + w = x. Notice that by defining n k fx = f i x. v = i=1 t f t f and flowing Σ 0 by v moves Σ 0 through the level sets of f. We need to make sure the speed of v is not too large. Since t f = f i f v i = f i f v ip + f i f v i v i p i i i and since the v i p are orthonormal and i f i = f we conclude 1 t f i v i v i p C r ε τ r on Ω r. By possibly shrinking ε depending on n, δ, M, we can control the right hand side by 1/ and thus get v on Ω r. Note that vi g j = 0 vi t g j = vi g j = vi gj n n = vi g j n n + g j n vi n = g j vi n n + g j n vi n and therefore vi t g j Cn Av j, C r ε τ r. Since v, we conclude that v t g j C r ε τ r 9

10 in Ω r and, consequently, that k+1 v t w Cn v g j t g j k+1 Cn v g j t g j + g j v t g j C r 3 ε τ r j=1 in Ω r, the extra r coming from the linear g j on the right. Let Ω r,f be the region obtained by integrating Σ 0 along v for time t r 3k, intersected with Ω r. By the fundamental theorem of calculus, as well as t w sup t w + C r sup v t w C ε φ r 0 + C r ε τ r Σ 0 Ω r,f j=1 w k C r ε φ r 0 + C r 5 ε τ r in Ω r,f. Since w is bounded from below, these bounds can be turned into C 1 bounds for w k. These in turn extend to C 1 bounds on the graph function u. 3.. Cylindrical estimates. In order to obtain the first Lojasiewicz inequality from Proposition.1 we need to absorb the A/H 1 term into a φ term. This is obtained by very generic estimates for surfaces that resemble cylinders in some sense. The key operators are the drift Laplacian L = 1 x t and L = L + A + 1. Recall that L is the linearization of M, the E-L functional of F. The second fundamental form and the function φ are related by a PDE: from which we get as a corollary that: L H A/H = φ + φa H LA = A + φ + φa + A φ + φ A H L H A/H = A/H + φ + φ A, A H + A φ + φ A H 3 where L f σ = Lσ + log f, σ for any tensor or function σ. Remark. One point is that every term involving A above is paired up with something linear in φ and we also have no derivatives of A showing up. The exception to both is the term A/H which thankfully has a favorable sign in the PDE. After integrating the second relationship by parts with suitable use of the Gaussian measure we obtain the following weighted L estimate: Corollary 1.. If Σ B R is smooth, H δ > 0 and A M, then for all s 0, R we can estimate A/H ] L Σ B R s C { 1 s volσ B R for C = Cn, δ, M. R } s + φ] L1 Σ B R + φ] L1 B R. By interpolation we can get L estimates from these L estimates provided we control a high enough derivative. Corollary 1.7. If Σ B R is smooth, H δ > 0, A + l A M for some l, and λσ λ 0, then for all y < R : A/H y + A/H y C R n{ φ] 1 c l,n L 1 B R + where C = Cn, λ 0, δ, l, M, and c l,n 0, 1 1 as l } c l,n R 1 y

11 3.5. Proof of First Lojasiewicz inequality. Theorem 0. - LI1. There exist ρ > 0 ρ = 5n + 10 and l 0 1 large depending on n, and a sequence c l,n 0, 1 1 as l such that if: then 1 Σ n R n+1 has λσ λ 0, l l 0, C l <, and 3 R < R ε0,l,c l graph, inf w C k] L C C Σ B R C Rρ{ φ] c l,n L 1 Σ B R + k } c l,n R for C = Cn, λ 0, ε 0, l, C l, where w C denotes the distance function to the axis of the cylinder C. Proof. Suppose we are given n, λ 0, ε 0, l, C l as above. Then by writing out the graphs and interpolation, A + A Mε 0, l, C l and H δε 0 on Σ B R. In fact φ 1,B5 n Cn, l φ L1,B 10 n + φ a l,n L 1 by interpolation. Choosing l 0 = l 0 n so that a l,n 3/ for all l l 0 gives seeing as to how we may suppose φ L1,B 10 n 1. We may suppose that the quantity: φ 1,B5 n Cn, l, C l φ 3/ L 1,B 10 n { R 1 sup r R : Cn, λ 0, δ, l, M R n+5 φ] 1 c l,n L 1 Σ B R + l φ 1 a l,n L,B 10 n ] 1 ] c l,n R 1 } r ε δ = δε 0, M = Mε 0, l, C l by writing out the graphs, and ε = ε n, δ, M as in.1 is larger than 5 n, C here being the constant in the interpolation corollary 1.7 plus the constant Cn, l, C l from a few lines above. If this is not true, we are immediately done since the constant C in the conclusion is allowed to depend on precisely the same parameters as the ones we have: n, λ 0, ε 0, l, C l. By 1.7 and the φ-estimate above we can bound for every 5 n r R 1 : r φ 1,B5 n + r 5 A/H 1,Br { C r φ 3/ L 1,B + r 5 R n φ] 1 c l,n 10 n L 1,B R + 1 ] c l,n R 1 } r C R n+5 φ] 1 c l,n L 1,B R + 1 ] c l,n R 1 r ε 1 and therefore by.1, Σ B r is a graph over some cylinder C, with u 1 the ression above prior to ε. We want to take r = R 1. Remark. In taking r = R 1 we get a crude term R 1/8 on the right hand side above that we cannot offset with the Gaussian measure. This is an ository issue in the way Proposition.1 is packaged. Instead one observes that the proof of.1 allows for us to have pointwise u + u bounds that depend on the distance to the origin. This way indeed we can pick r = R 1 and conclude that: uy + uy C R n+5 φ] 1 c l,n L 1,B R + 1 ] c l,n R 1 y for y B R1. 11

12 At this point we re done because by the bound on uy in the remark above u] L,B R1 = uy y dv y B R1 ] C volb R1 R n+10 φ] c l,n L 1,B R + c l,n R ] C R ρn φ] c l,n L 1,B R + c l,n R w ] k] L,B R1 C R ρn φ] c l,n L 1,B R + c l,n R and on B R \ B R1 we have the crude estimate w k x, so that w k] L,B R \B R1 C R n+ R 1 C R 5n+1 φ] c l,n L 1,B R + ] c l,n R the last inequality following from the definition of R 1. The result follows by adding these two inequalities.. Second Lojasiewicz Inequality The second Lojasiewicz inequality captures the proximity of the F -value of a surface to that of a critical point of the F functional in terms of the gradient φ = Σ F, and a noncompact error term with onential decay. To prove this, we will: 1 Establish elliptic estimates for the operator L and relevant Sobolev estimates on ker L and ker L. Using those estimates and by looking at what happens near ker L versus near ker L we will establish Proposition.1, which bounds the same quantity as LI F Σ F C in terms of φ, a noncompact error term with onential decay, and also a measure of L -closeness to a cylinder. 3 We will then swallow the L -closeness term into the prior two terms in view of LI1..1. Understanding the kernel of L. In this entire subsection Σ refers to a round cylinder in C k, in which case L collapses to L + 1, seeing as to how A 1/. Furthermore, all L p and Sobolev norms are weighted by the Gauss measure. The following lemma summarizes all the tools we have available for us from functional analysis: Lemma 3.. The following are true for L: 1 It is self-adjoint on H, with u Lu Σ x = u, v Σ x. The embedding H 1 L is compact. 3 The operator L has discrete spectrum with finite multiplicity in H. L admits a complete orthonormal basis of C functions. We can also show Lemma 3.. There exists C depending on n such that for all u H 1. x u] L C u] L + R n ku] L C u] H 1 Remark. There is nothing surprising here. The inequality on the right is trivial, and the inequality on the left says that the x weight can be swallowed into the gradient integral on the right, in fact even just the gradient in the flat directions. 1

13 Sketch of proof of 3.. We provide the sketch of this proof because really all weighted Sobolev norms proofs in this paper are more or less the same, so it s good to outline a few of them. Without loss of generality u has compact support. If we write y = x t for the R n k factor coordinates, then x = y + k on any cylinder in C k it suffices to bound y u] L instead of x u] L. We compute div Σ u y x x = u u, y + n k u u y The result follows after integrating. R n ku + n k u u y The following lemma provides rudimentary elliptic estimates for the projections u K, u on K = ker L and ker L s taken in L. Lemma There exist C > 0 large, µ > 0 small depending on n, k such that µ u ] H Lu] L C u] H u K ] H C u K ] L Proof. The inequality Lu] L C u] H is a corollary of 3.. The inequality µ u ] H Lu] L is slightly stronger than the standard elliptic inequality which would give an H 1 estimate, not H, so let s see how to prove that. We claim that u] H C u] L + Lu] L Notice that this is standard for H 1 on the right integrate by parts and use self adjointness of L, so we really just need to show the same upper bound for u. ] } div Σ { u u Lu u x x } { } = div Σ { u u div Σ Lu u u u Lu u, 1 xt { 1 { } = u } Lu, u + Lu u { 1 x t u, u Lu 1 } x tu 1 = u Lu 1 xt u, u Lu, u = 1 u Lu 1 x t 1 u Lu, u 1 = u Lu 1 xt u, u Lu, u 1 = u Lu 1 xtu, u Lu, u = u + Ric u, u + u, u Lu 1 xt u, u Lu, u = u + Ric u, u Lu u Lu where we ve used in the following order the facts that: 1 x t u = x tu on the cylinder, the Bochner formula 1 u = u + Ric u, u + u, u, and 3 Ric 0 on the cylinder in the last four steps. Integrating, we get u] L Lu] L as claimed. By applying this elliptic estimate to u, ] u ] H C u ] L + Lu ] L ] C u ] L + Lu] L C Lu] L, 13 ]. ] = C u ] L + Lu u ] L

14 the last step holding true by ellipticity. For the last claim observe that Lu K = 0 Lu K = u K, and that by the previous application of Bochner s theorem this forces u] L Lu] L = u K ] L. This bounds the second derivative. For the first derivative, integrating Lu K = u K by parts gives u K ] L = u K ] L. The result follows. Next we seek to understand K = ker L. In fact one can compute exactly what elements make up the kernel: Lemma 3.5. Each v K can be written as vy, θ = qy + i y i f i θ + c, where q is a homogeneous quadratic polynomial on the R n k factor, f i is an eigenfunction on S k k with eigenvalue 1/, and c R. Proof. This essentially follows from the fact that square-integrable with respect to the weighted measure L-harmonic functions are constant. Indeed suppose f C Σ L Σ be such that Lf = 0. If ζ is an arbitrary test function then Lf = 0 ζ f, Lf = 0 ζ f, f = 0 ζ f] L = f ζ, f. For appropriate cutoff functions ζ, ζ C ζ, so ζ f] L C ζ f f ] L 1 1 ζ f] L + C f] L, i.e., ζ f] L C f] L. Since ζ were arbitrary, f H 1. Repeating this for the L L-harmonic function f we deduce that f H so Lf L. Therefore f] L = f, Lf = 0, i.e. f is constant. The rest of the argument is clearly described in the paper. Corollary There exists C depending on n such that for any v K, then v C 1 + y v] L, v C 1 + y v] L, v, R n k C v] L, v C 1 + y v] L. Remark. This result is important because it says that the only Jacobi fields with respect to e x / dv x i.e. with onential growth actually never grow faster than quadratically. By treating the quadratic growth as the product of two linear growths, in combination with the previous Poincare-type lemma we get: Lemma 3.. There exists C depending on n such that for any u H, {u K } C u K ] L and {u } C u u ] H where the quantity {u} is defined to be {u} u + u + u, R n k x 1 u ] L Σ Remark. Observe that {av} = a {v}. One should think of {u} as a weighted W, norm. Therefore the lemma using the corollary controls the weighted W, norm of elements of K by the L norm... Preliminary gradient estimate. Observe that since L is the linearization of Σ F, one can prove the following standard nonlinear estimate: Lemma.3. There exist ε > 0 small and C > 0 large depending on n such that if u : C R has u,σ ε, then Mu Lu] L Σ C {u} and F u F Ck 1 u, Lu C u]l {u} Sketch of proof. One can compute M to be of the form Mup = fu, u + p, V u, u + Φu, u, u where f, V, Φ depend smoothly on s, y for s small. Since L is the linearization of M and cylinders are a critical point for M, we have the standard nonlinear pointwise estimate Mu Lu C 1 + x u + u + C u + u u where C = Cn. The 1 + x term shows up because of the naked p in the ression of Mu. Integrating in space and using Young s inequality, Mu Lu] L C 1 + x u ] L + C 1 + x u ] L + C 1 + x 1 u ] L 1

15 The first claim follows by Lemma 3.. The second claim follows by integration. Proposition.1. There exists C > 0 large depending on n, λ 0, and ε > 0 small depending on n such that if: 1 λσ λ 0, Σ B R is the graph of some function u over C B R, with u,σ BR ε, and 3 β 0, 1, then F Σ F C C φ] 3+β L Σ B R + C Rn β R 3+β 1+β + C u] L Σ B R. Remark. The first two terms on the right are important when u K is tiny, and the last term is important when u is somewhat small. Proof of.1. First we may extend u to a function defined on the entire cylinder by multiplying with an appropriate cutoff function. If Σ u denotes the graph of this entire function u then we have F Σ F Σ u Cn, γ λ 0 R n 1 γ R for γ 0, 1 which we can just take to be 3+β < γ < 1. Therefore we might as focus on bounding F 0 u F Σ u F C. The point is to use the nonlinear comparison inequalities we ve come up with: 1 Mu] L Lu] L C {u}, and F 0 u 1 u, Lu C u] L {u}. In the second item above we ve implicitly used that u, Lu = u, Lu. We also need to make the most that we can out of the crude estimate Mu] C φ] L + R. 8 Indeed: F 0 u 1 u, Lu + C u] L {u} 1 u ] L Lu ] L + C u] L {u K } + C u] L {u } C u] L u ] H + C u] L {u K } + ε C u] L u ] H C u] L u ] H + C u] L {u K } where we ve used Lu ] u ] H and {u } C u,σ u ] H. Here C = Cn. Now there are two cases to consider: Suppose that the kernel component of u is tiny, i.e. u is essentially normal to the kernel. Then we can essentially control everything with a power of u ] H that is sufficiently large to make up for our crude Mu] bound. In particular, suppose {u K } η u ] 1+β H Then {u} {u K } + {u } η u ] H + C u,σ u ] H 1 µ u ] H as long as we require η + C ε 1 C 1 µ, a quantity depending on n and C = Cn being the constant below. Then Mu] L Lu] L C {u} 1 µ u ] H Since u] L u K ] L + u ] L C {u K } + u ] H, we can continue our F 0 u estimate as follows: F 0 u C u] L u ] H + C u] L {u K } C {u K } + u ] H u ] H + C {u K } + u ] H {uk } C {u K } 3 + C {u K } u ] H + C {u K } u ] H + C u ] H C η u ] 31+β + C u ] 3+β + C u ] H C u ] 3+β 15

16 because the middle term dominates for small u. But now we re done because F 0 u C u ] 3+β C µ 1 Mu] 3+β L C φ] 3+β L + C 3 + β R The second case to consider is that of the component normal to the kernel being somewhat small. Then since we have excellent estimates near the kernel, we are in good shape. That is, if Then and the proposition follows. {u K } η u ] 1+β H F 0 u C u] L u ] H + C u] L {u K } C η 1 u] L {u K } 1+β C u] L u] 1+β L 3+β 1+β = C u] L Notice that we ve been using ε instead of ε 0 in this subsection. This is because ε 0 is required to be much smaller to allow for some wiggle room to allow us to jump between cylindrical graphs: Lemma.9. There exists ε 0 > 0 small depending on ε, n, such that if a surface Σ is 1 an R 1 -graph over a cylinder C 1 with 1,BR1 ε 0, and an R -graph over another cylinder C with,α,br with 5 n R 1 < R, then it is also an R-graph, R = min{r 1, R }, over C 1 with,br ε. Remark. This and LI1 on spheres are the two lemmas that determine the value of ε 0 that is to be used throughout the paper. Remark. Why is this lemma important? Because the graphical radius does not have a fixed cylinder associated with it, while quantitative results like Proposition.1 provide concrete graphical estimates on a particular cylinder that may or may not match that of the graphical radius. We re going to have to transplant the graphical cylinder into the quantitative cylinder..3. Proof of Second Lojasiewicz inequality. Recall the statement of LI: Theorem LI. There exists ρ > 0 and l 0 1 large depending on n, and a sequence c l,n 0, 1 1 as l such that if: then 1 Σ n R n+1 has λσ λ 0, l l 0, C l <, 3 R < R ε0,l,c l graph, and β 0, 1, { F Σ F C k C R ρ φ] c 3+β l,n 1+β L Σ B R + c l,n for C = Cn, λ 0, ε 0, l, C l. Proof. Arguing as in LI1, we may assume that R 1 sup { r R : R n+5 φ] 1 c l,n L Σ B R β 1 + β R β 1 ] c l,n R 1 } r η } R is large enough that R 1 > 5 n, for a choice of η that allows us to employ Proposition.1. Accordinng to.1, there exists a cylinder C C k over which Σ B R1 is a graph of a function u with u 1,C BR1 C R ρ φ] 1 c l,n L,B R1 + 1 ] c l,n R 1 R 1 ε 0 16

17 by a smaller choice of η > 0 and by LI1, u] L Σ B R1 C Rρ φ] c l,n L Σ B R + ] c l,n R By the rotation lemma,.9, Σ is also a graph over C B R, R = min{r 1, R}, with u,c BR ε. Notice that the LI1 bound can be extended from B R1 to B R since u] L Σ B R \B R1 C ε R n R 1 ] C ε R ρ φ] c l,n L Σ B R + c l,n R the latter by the very definition of R 1. That is, u,σ BR ε and where C = Cn, λ 0, ε 0, l, C l. By.1 we can estimate u] L Σ B R C Rρ φ] c l,n L Σ B R + F Σ F C C φ] 3+β L Σ B R + C Rn β ] c l,n R, R 3+β 1+β + C u] L Σ B R. The φ] term is going to be dominated by a smaller power of φ] soon, so we won t worry about it. The last term is bounded by LI1 in a favorable way. As for the middle term, if R = R then we are done because the bound is favorable. If R < R then R = R 1, so 3 + β R = 3 + β R1 ] c l,n 3 + β R all of which are favorable. Combining all the dominant terms in these estimates we get LI. = C R ρ φ] c l,n 3+β L Σ B R + 5. Scale Comparison Theorem Now we discuss what is essentially the final and most important building block of the proof: the scale comparison theorem Theorem 5.3. The key tool in proving this is going to be Brian White s local curvature estimate for MCF. We will use an improvement and extension argument to get good curvature estimates forward in time, and then drag those backwards. The plan of action is an improvement/extension scheme: 1 As long as we are below the shrinker scale and we have crude bounds on some large ball B R, we can find strong C,α estimates on a slightly smaller ball B 1 τr. This is called the improvement scheme. As long as we have strong C,α bounds on a ball B R, we can find crude estimates on a larger ball B 1+θR. This is called the extension scheme. Remark. The statements in this section of the companion particularly in the extension section differ somewhat from those of the paper. The reason is that I decided to keep very licit track of the constants, and that required some rephrasing here and there. The statements here make more sense to me. We will assume throughout that we are working on the RMCF time interval 1, 1] Improvement argument. The first step in the improvement argument is an extension of LI1 which improves the C 1 estimate obtained in Proposition.1 into a C,α estimate using interpolation. Theorem.5. There exists l 0 1 large depending only on n, and there exist R 0 > 0 large and σ 0 > 0 small depending on n, λ 0, δ 0, µ, l l 0, M such that if: 1 Σ n R n+1 has λσ λ 0, Σ B R has A + l A M and H µ > 0 for some R > R 0, l l 0, and 3 Σ B R0 is C σ 0 -close to a cylinder in C k, 17

18 then Σ B R1 is C,α δ 0 -close to a possibly different cylinder in C k, where { R 1 sup r R : R n+5 φ] 1 c l,n L 1 Σ B R + 1 ] c l,n R 1 } r γ and where γ = γn, λ 0, µ, l, M. Proof. We mimic the proof of LI1 for the most part. As before we have φ 1,B5 n Cn, l φ L1,B 10 n + φ a l,n L 1,B l φ 1 a l,n 10 n L,B 10 n ] and so we continue to choose l 0 = l 0 n 1 large enough so that a l,n 3/ for all l l 0, because then φ 1,B5 n Cn, l, M φ 3/ L 1,B. We may suppose that 10 n { R 1 sup r R : Cn, λ 0, µ, l, M R n+5 φ] 1 c l,n L 1 Σ B R + 1 ] c l,n R 1 } r ε is 5 n. Here ε is actually ε = ε n, µ, M as in.1, and also squeezed further depending on n, λ 0, µ, l, M, δ 0 in a way that will be determined at the end of the proof. The C is the constant of 1.7 plus the constant Cn, l, M above. This assumption is valid because we may squeeze the negative R onential to be small by making R 0 large and φ] L 1 to be small by making σ 0 small depeding on n, λ 0, µ, l, M. Of course we also need to have σ 0 the ε 1 n, µ, M of.1. At this point we may apply.1 like we did before to get that Σ B R1 is C 1 close to a cylinder, with 1,BR1 Cn, λ 0, µ, l, M R n+5 φ] 1 c l,n L 1 Σ B R + 1 ] c l,n R 1 R 1 In other words we can control the C 1 norm of the graph function on B R1, but also the l + -nd norm in view of l A M on B R, R R 1. By interpolation,,α,br1 η l+,br1 + Cη, l 1,BR1 η Cn, M + Cη, l ε Pick η > 0 small enough depending on n, M such that the first term is δ 0 /. Now the choice of ε is clear: pick it depending on η, l, δ so that the second term is δ 0 /. Claim Improvement step. There exist l 0 1 large depending on n, τ, and R 0 > 0 large depending on n, λ 0, ε 0, δ 0, K, τ, l, C l such that if: 1 Σ s is RMCF with λσ s λ 0 for s 1, 1], l l 0, C l <, 3 s 0 1, 1], τ 0, 1/], R R 0, R R entr Σ 0, R R ε0,l,c l graph Σ s 0, and 5 Σs0 F B R K F Σ 1 F Σ 1, then Σ s0 B 1 τr is C,α δ-close to a cylinder. Proof. Observe that ] φ] c l,n L Σ s0 B R = Σ s0 F c l,n B R K 1 1 c l,n F Σ 1 F Σ 1 c l,n K c l,n RentrΣ 0 The key quantity in Theorem.5 at r = 1 τ R is bounded from above by ] c l,n R 1 τ R K R ρ c l,n RentrΣ t ] + c l,n R K R ρ 1 τ c l,n R K τ R 0 R ρ φ] c l,n L Σ s0 B R τ R

19 since R R entr Σ 0, and since we may require l 0 to be larger yet depending on n, τ so that c l,n 1 τ for all l l 0. Additionally require l 0 to exceed the l 0 in Theorem.5. Then, requiring in addition that R 0 be large depending on all the parameters listed in the statement and the fixed choice of l 0, Theorem.5 kicks in to give the required C,α estimate. 5.. Extension argument. Key in the extension argument is the following result due to Brian White: Claim. There exist σ = σn > 0 and C = Cn > 0 such that if: then for all s τ/, 0] we can estimate: Here s M s is a MCF, not RMCF. πτ M n/ x x 0 dv x 1 + σ τ τ A C τ 1 on M s B τ/x 0. Remark. This is not the way the theorem was stated in Brian White s paper, but rather Toby Colding and Bill Minicozzi s statement. It was not immediately clear to me that Brian White s theorem implies this, but Brian thinks it s fine. Note, by the way, that the symbol τ plays an entirely different role here in the extension section than it did in the improvement section. Lemma There exists δ 0 > 0 small, θ > 0 small depending on n, ε 0, M, such that if: 1 R > n, M τ is MCF such that A 3,Mτ B R+ M for τ 1 1/M, 1 + 1/M], and 3 M 1 is C,α δ 0 -close to a cylinder C C k, then M τ B R is C,α ε 0 -close to τ C for τ 1 θ, 1 + θ]. Remark. The importance of this lemma is clearer when it is reformulated in terms of RMCF. If 1 Σ t is RMCF with curvature bounds on t 1/M, t + 1/M], within B R, that depend on M, and Σ t B R is C,α δ 0 -close to C C k, 3 R R 0, then Σ t+η B 1+κR is C,α ε 0 -close to the same cylinder, for η, κ, δ 0, R 0 depending on ε, ε 0, M. It is certainly not surprising that the radius gets larger as time moves forward, since our reference frame is a shrinking cylinder of the original flow. Another important ingredient is the following mean-value type estimate: Lemma 5.3. There exists a constant C > 0 large depending on M such that if: then 1 Σ t is RMCF on t 1, t ], β 0, t t 1, R > 0, and 3 A M on Σ s B R+1, for all s t 1, t ], Proof. One may compute that max s t 1+β,t ] φ] L Σ s B R C + β 1 F Σ t1 F Σ t t L φ = φ A + 1 φ If ζ is a smooth cutoff function between B R B R+1, with 0 ζ 1, ζ, then by taking the manifold movement into consideration, we see: t φ ζ = φ 3 ζ, n + ζ Lφ + φ A + 1 φ. 19

20 Then by direct computation we can estimate t φ Σ ζ x dv x = t φ ζ x dv x Σ t + A + 1 φ ζ x Σ t φ ζ x dv x Σ t + φ 3 ζ, n x dv x Σ t φ, ζ x dv x Σ t A + 1 ζ + 5 ζ φ Σ t φ ζ Σ t x dv x dv x x dv x The last term may be dropped we need to keep it to get a spacetime φ L -estimate. In any case, for any s t 1 + β, t ]: x dv x as claimed. φ ζ Σ s min φ η x t t 1,t 1+β] Σ t t C + β 1 φ Σ t t 1 x t dv x + C t 1 dv x Σ t φ x dv x Remark. This is not quite the way the lemma is stated in the paper. It was more convenient for me to state it this way. Also, I have not stated one of the estimates of the lemma that I m not going to use anywhere. Another important ingredient is the following result: Corollary There exists R 0 = R 0 n, λ 0, σ, τ > 0 such that if: 1 Σ s is RMCF with λσ s λ 0, s 1, 1], R R 0 +, R R entr Σ 0, x 0 B R R0, τ 0, 1/], and 3 we can control πτ Σ n/ x x 0 dv x 1 + σ 1 τ then we can also control for all s 1, 1]. πτ Σ n/ x x 0 dv x 1 + σ s τ Proof. In view of the entropy bound there exists r 0 = r 0 n, λ 0, σ > 0 uniform over s 1, 1], y R n+1 such that πτ n/ x y dv x σ τ Σ s\b r0 τ y 0

21 By use of a cutoff function, it s relatively simple to show that x x 0 dv x Σ s B R τ x x 0 dv x + R + 1 φ x x 0 + φ dv x dt τ τ 1 Σ t B R+ τ Σ 1 B R+ The second term on the last line can be estimated by Cauchy-Schwarz and our entropy bounds: R + 1 φ x x 0 dv x τ 1 Σ t B R+ τ R + R + 1 ] 1/ πτ τ n/ λ 0 φ] L 8 Σ dt t 1 R + R + πτ τ n/ λ 0 R entrσ 0 µn, λ 0, R 0 τ 1 8 where µn, λ 0, R 0 can be made arbitrarily small by requiring that R 0 be large depending on n, λ 0, and the smallness factor. Along the same lines, 1 φ dv x 1 Σ t B R+ R + 1 R + φ] L Σ t dt R entrσ 0 µ R 0 1 which can once again be made arbitrarily small by choosing R 0 large. Putting it altogether, for s 1, 1] and x 0 B R R0 arbitrary we can estimate: πτ Σ n/ x x 0 dv x s τ = πτ Σ n/ x x 0 dv x + πτ n/ x x 0 dv x s B R τ Σ s\b R τ πτ Σ n/ x x 0 dv x + πτ n/ µn, λ 0, R 0 τ 1 + πτ n/ µ R 0 + σ 1 τ 1 + 3σ + πτ n/ µn, λ 0, R 0 τ 1 + πτ n/ µ R 0 This can be made 1 + σ by choosing R 0 = R 0 n, λ 0, σ, τ sufficiently large. Remark. Once again, this is not the way the result is stated in the paper. The second important lemma makes use of Brian White s local recularity theorem. Proposition 5.6. There exists δ 0 > 0 small depending on n, and there exists R 0 > 0 large depending on n, λ 0, τ, and also C l > 0 large depending on n, l such that if: 1 Σ t is RMCF with λσ s λ 0, s 1, 1], R R 0 +, R R entr Σ 0, x 0 B R R0, τ 0, 1/], and 3 we can estimate A δ 0 τ 1 on Σ 1 B R0 τ x 0 then for all t 1, 1], s t log1 7τ/8, t log1 τ] 1, 1], we can further estimate: A + τ l l A C l τ on Σ s B τ/3e 1 s t x 0. Sketch of proof. In view of the entropy bound there exists R 0 > 0 uniform over s 1, 1], y R n+1 such that πτ n/ x y dv x σ τ Σ s\b R0 τ y 1

22 where σ is as in Brian White s local regularity theorem referred to as ε there. If one picks x 0 B R R0 and dilates by τ 1/ around that center point then the curvature bound becomes: A δ 0 on τ 1/ Σ 1 x 0 B R0 Provided δ 0 is small enough depending on σ, the region τ 1/ Σ 1 x 0 B R0 = τ 1/ Σ 1 B R0 τ x 0 x 0 is going to look essentially planar centered at the origin and the non-compact contribution of the onential integral is small; i.e., πτ Σ n/ x x 0 dv x 1 + σ 1 τ. Since we re now in the setup of Corollary 5.15, it is also true that πτ Σ n/ x x 0 dv x 1 + σ s τ holds true for x 0 B R R0, s 1, 1], arbitrary since R R 0 +. Since the density bound is true at time s 1, 1], by looking at the corresponding MCF Brian White s local regularity theorem gives a crude curvature bound forward in MCF-time on each ball B τ/x 0, x 0 as above. Translating that back to RMCF, we need to keep track of the fact that an MCF-static x 0 moves under RMCF, so instead the bound we got was the one claimed. The τ/ radius shrinks to a τ/3 radius once we use Shi-like estimates to bound l A given the A bound. Remark. Observe that in view of the latter estimate being over the moving point e 1 s t x 0 in t log1 7τ/8, t log1 τ] 1, 1], Proposition 5.6 gives us curvature bounds on a multiplicatively larger ball. Claim Extension step. There exist R 0 > 0 large, δ 0 > 0 small, θ > 0 small, K > 0 large depending on n, λ 0, ε 0, and there exists C l > 0 large depending on n, λ 0, ε 0, l such that if: 1 R R 0 +, R R entr Σ t, Σ s B R is C,α δ 0 -close to a cylinder depending on s for each s 1/, 1], then for every s 1/, 1]: R ε0,l,c l graph Σ s 1 + θ R and Σs F L B 1+θR K F Σ 1 F Σ 1 Proof. Let δ 0 = δ 0 n be as in Proposition 5.6. Let τ = τn, δ 0 > 0 be such that A δ 0 /τ for each point in Σ s B R, s 1/, 1]. This can be done seeing as to how our surfaces are C,α δ 0 -close to a cylinder at each time. By 5.6 we may estimate A + τ l l A C l τ on Σ s B τ/3e 1 s t x 0 for all x 0 B R R0, R 0 = R 0 n, λ 0, τ, and all s t log1 7τ/8, t log1 τ] 1, 1], and all t 1, 1]. Said otherwise, we have curvature and curvature l-th order derivative bounds on a ball B 1+κR on a time interval s 1 log1 7τ/8, 1]. The crude curvature bounds immediately give the required estimate: max s 3/,1] φ] L Σ s B 1+κR K F Σ 1 F Σ 1 which implies the second inequality claimed in the statement. 1 Also, max s 3/,1] φ] L Σ s B 1+κR K K By interpolation, φ,α,σs B 1+κR R entrσ 0 R. C φ L Σ s B 1+κR + C φ l,σs B 1+κR R C φ] L Σ s B 1+κR + CC 3, τ CK, R 0, C 3, τ = Cn, λ 0, ε 0, δ 0, C 3 1 The remainder of the proof can be simplified after a clarification kindly offered by Bill Minicozzi and Toby Colding.

23 In view of these crude bounds on the speed of evolution in the RMCF there exists ξ = ξn, λ 0, ε 0, δ 0, C 3 such that we can go back in time by from 1/ to 1/ ξ and have our C,α δ 0 -bounds on B R become C,α δ 0 -bounds at worse, on B R still except earlier in time. If δ 0 = δ 0 n, ε 0, C 3 > 0 is small enough, by the uniform stability of cylinder lemma, 5.39, these extend to C,α ε 0 -bounds on a ball B 1+θR slightly forward in time from the instant t = 1/ ξ, say starting at 1/, where θ = θn, ε 0, C 3, ξ can be chosen small enough that θ σ. In summary, there exists θ = θn, λ 0, ε 0 > 0 such that we have C,α ε 0 -tight bounds on B 1+θR on the original time interval 1/, 1], and curvature bounds A + τ l l A C l τ. By replacing C l by τ l 1 C l 1/, this implies that R ε0,l,c l graph 1 + θ R as claimed Proof of Scale comparison theorem. We remind the reader that the theorem read: Theorem 5.3. There exist µ > 0 small, l 0 1 large, C > 0 large depending on n, λ 0, ε 0, such that if: 1 Σ t is a RMCF on 1, 1], and l l 0, then there exists C l > 0 large depending on n, λ 0, ε 0, l, such that 1 + µr entr Σ 0 min 1/ t 1 Rε0,l,C l graph Σ t + C Proof. Let R 0, δ, θ, K be as in the extension step, depending on n, λ 0, ε 0. Let τ = θ/ be the parameter that is to be used in the improvement step. Let l 0 1 be as in the improvement step, depending on n, τ. Let C l0 > 0 be as in the extension step, depending on n, λ 0, ε 0, λ 0. Finally, let R 0 > 0 be as in the improvement step depending on n, λ 0, ε 0, δ, K, l 0, C l0, quantities that have already been determined. The dependence of everything so far traces back down to n, λ 0, ε 0. Notice that if the minimum graphical radius ρ min s t 1/,t+1] Rε0,l0,C l0 graph Σ s is smaller than 1 τ 1 R 0 there is nothing to show: we may take C = R 0. So let s suppose that ρ 1 τ 1 R 0, and of course that it s R entr Σ t. Then by the improvement step, Σ s B 1 τρ is C,α close to a cylinder, for every s t 1/, t + 1]. By the extension step, min Σ s 1 + θ1 τρ We may iterate this until s t 1/,t+1] Rε0,l0,C l0 graph min s t 1/,t+1] Rε0,l0,C l0 graph Σ s 1 + θ1 τρ At which time we may apply the extension step once more to get a multiplicatively better factor. The result follows for l = l 0, and in particular we have curvature bounds all the way up to the entropy radius. This lets us apply the same argument for any l l 0. 3