Singular Behavior of Minimal Surfaces and Mean Curvature Flow

Stephen James Kleene

A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy.

Baltimore, Maryland

April, 2010

c Stephen James Kleene 2010

This document records three distinct theorems that the author proved, along with his collaborators, while a graduate student at The Johns Hopkins University. The author, with M. Calle and J. Kramer, generalized a sharp estimate of Tobias Colding and William Minicozzi for the extinction time of of convex hyper-surfaces in euclidean space moving my their mean curvature vector to a much broader class of evolutions studied by Ben Andrews in (1). Also, the author gave an alternate proof, ﬁrst given by D. Hoﬀman and B. White in (17), of very poor limiting behavior for sequences of minimal surfaces in euclidean three space. Finally, the author, together with N. Moller in (23) constructed a new family o asymptotically f conical ends that satisfy the mean curvature self shrinking equation in euclidean three space of all dimensions.

ii Acknowledgements

Thanks Bill Minicozzi for his continuing advice and support during my time as his student. Thanks to Bill, Chika Mese, and Joel Spruck for the many excellent classes on Geometry and PDE’s over the years; they have beneﬁted me immeasurably. Thanks to all my wonderful friends here in Baltimore, who provided vital advice and support in all things non-mathematical. Thanks to Morgan and Nevena, who always gave me a place to stay and food to eat (and cash) when I needed it. Thanks to Mom, Dad, Julia, Morgan and Granny for everything else, too numerous to tabulate.

iii Contents

Abstract ii

Acknowledgements iii

List of Figures vi

Introduction 1

1 Introduction 1 1.1 OverviewofResults...... 1

2 Background Material 8 2.1 The Volume Functional, Minimal Surfaces and the Weierstrass Repre- sentation...... 8 2.2 MeanCurvatureFlow ...... 11 2.3 Singularities of Mean Curvature Flow and the Equation for Self Shrinkers 13 2.4 SweepoutsandthenotionofWidth...... 13

3 Width and Mean Curvature Flow 15 3.1 Contraction of Convex Hypersurfaces in Rn+1 ...... 16 3.2 WidthandtheexistenceofGoodSweepouts ...... 17 3.3 Fconvex...... 19 3.4 Fconcave ...... 20 3.5 ExtinctionTime...... 21 3.6 Fhomogeneousofdegreek...... 23 3.7 2-Widthand2-ConvexManifolds ...... 25

4 A Singular Lamination 28 4.1 TheStructureoftheProof...... 29 4.2 deﬁnitions...... 31 4.3 PreliminaryResults...... 32 4.4 Proofoflemma18 ...... 34

iv 4.5 ProofofTheorem17(A),(B)and(C) ...... 38 4.6 Proof of Theorem 17 (D) and The Structure Of The Limit Lamination 39 4.7 Appendix ...... 41

5 Self-Shrinkers with a symmetry 44 5.1 Non-compact, rotationally symmetric, embedded, self-shrinking, asymptotically conical ends with positive mean curvature ...... 44 5.2 Initial conditions “at inﬁnity” for a non-linear ODE with cubic gradient term ...... 45

Bibliography 51

v List of Figures

4.1 The functions Fk ...... 30 4.2 A schematic rendering of the domain Ωk in the case of M = pl = l { 2− l N ...... 32 − | ∈ } 4.3 The“goodintervalaboutp”...... 37

vi To my family Because I love them so...

vii Chapter 1

Introduction

My research has been in the fields of minimal surfaces and mean curvature flow. I have three distinct results in these fields, recorded in (8), (24), and (23). In (24), I studied laminations in R3 by minimal surfaces, advancing the understanding of the famous structure theorem for minimal laminations of Colding and Minicozzi given in (12). In (8), with M. Calle and J. Kramer, I generalized a well known extinction estimate of Colding and Minicozzi for convex hypersurfaces moving by mean curvature to a very general class of geometric evolutions. Very recently, with N. Moller, I constructed a new family of rotationally symmetric surfaces that shrink homothetically under mean curvature flow, recorded in (23).

1.1 Overview of Results

In (24), I constructed a sequence of smooth minimal surfaces, embedded in a ball in R3, with very poor limiting behavior along a line segment. Generally one can consider a sequence of minimal surfaces Σ contained in balls B with B B k k k ⊂ k+1 in a manifold M. Setting B = B , and deﬁning K B to be the subset of B ∪k k ⊂ where the curvature of Σk blows up on a subsequence, one sees that (after passing to a subsequence) Σ converges to a smooth minimal lamination of B K. It is natural k \ then to try to understand the structure of the set K and to understand when the lamination extends smoothly across K. In several cases, this is well understood.

1 In the case that M = R3 and B = R3, Colding and Minicozzi proved that if K is non empty then it is a Lipshitz curve and the lamination is a foliation by parallel planes. In particular, the lamination extends smoothly across K. 3 In the case that M = R and the sets Bk are balls centered at the origin with uniformly bounded radii, the limit lamination need not be so well behaved. In particular, Colding and Minicozzi constructed a sequence of properly embedded minimal disks in a fixed ball in R3 that converge to a non-proper limit lamination of B x -axis that \{ 3 } extends smoothly at every point away from the origin. Similar results were obtained by B. Dean and S. Khan in (14) and (22), respectively. D. Hoffman and B. White in (17) gave the first example with Cantor set singularities and of singular sets with non-integer Hausdorff dimension. Inspired by this result, I proved the same, but with different methods:

Theorem 1.1.1. Let M be a compact subset of x = x = 0, x < 1/2 and let { 1 2 | 3| } C = x2 +x2 =1, x < 1/2 . Then there is a sequence of compact embedded minimal { 1 2 | 3| } disks Σ C with ∂Σ ∂C and containing the vertical segment (0, 0, t) t < 1/2 k ⊂ k ⊂ { || | } so:

2 (A) limk AΣ (p)= for all p M. →∞| k | ∞ ∈ 2 (B) For any δ > 0 it holds supksupΣ M AΣ < where Mδ = p M Bδ(p). k\ δ | k | ∞ ∪ ∈ (C) Σ x axis = Σ Σ for multi-vauled graphs Σ , Σ . k \{ 3 − } 1,k ∪ 2,k 1,k 2,k (D) For each interval I =(t , t ) in the compliment of M in the x -axis, Σ t < 1 2 3 k ∩{ 1 x < t converges to an imbedded minimal disk Σ with Σ¯ Σ = C x = 3 2} I I \ I ∩{ 3 t , t . Moreover, Σ x axis = Σ Σ , for multi-valued graphs Σ and 1 2} I \{ 3 − } 1,I ∪ 2,I 1,I Σ each of which spirals in to the planes x = t from above and x = t 2,I { 3 1} { 3 2} from below.

Hoﬀman and White had used a variational approach in their paper, which has the advantage of being extremely ﬂexible. However I followed (13) and (22) and constructed the sequence with explicit choices of Weierstrass data for each immersion

2 which, while more complicated in the technicalities, is advantageous for the explicit- ness of the construction. In a slightly diﬀerent direction, Meeks and Weber found sequences of minimal surfaces converging to a minimal lamination with singular set equal to any C1 curve (indeed in the local case, the singular set must lie on a C1 curve). Nonetheless, it remains an open question as to whether Cantor set like singularities can occur along curves that are not line segments. Moreover, we have the following conjecture that generalizes Theorem 1.1.1:

Conjecture 1.1.2. Any compact subset of a geodesic segment in a three manifold is the singular limit of a sequence of embedded minimal disks, if the geodesic segment is short enough

In (23), N. Moller and I proved the following:

Theorem 1.1.3. In R3 there exists a 1-parameter family of rotationally symmetric, embedded, self-shrinking, asymptotically conical ends Σ2 with positive mean curva- 1 ture, each being smooth with boundary ∂Σ isometric to a scaled S in the x2x3-plane centered at the origin. In fact for each symmetric cone in x 0 R3 with tip at the origin, there is C { 1 ≥ } ⊆ a unique such a self-shrinker, lying outside of , which is asymptotic to as x . C C 1 →∞ Most solutions for mean curvature flow Σ become singular in finite time, and { t} thus cease to exist in a classical sense. In (19), Huisken showed that as long as the principal curvatures remain bounded, the flow remains smooth, hence the existence of a singularity implies curvature blowup for the family Σ . Also in (18), it was { t} 1/2 shown that if the maximum curvature blows up no faster than c(T t)− , where − T denotes the singular time for the flow, then the asymptotic shape of the solution will be that of a self-shrinking shrinker. As shown in (6), the study of rotationally symmetric self-shrinking surfaces reduces to the study of the following system in the

3 x r plane: − x˙ = cos θ r˙ = sin θ x n 1 r θ˙ = sin θ + − cos θ 2 r − 2 where θ is the angle the tangent vector makes with the x-axis. When rotated about the x -axis in Rn (identiﬁed with the x-axis of the x r-plane), the resulting surface of 1 − revolution shrinks by homotheties under mean curvature ﬂow. For solutions graphical over the x-axis this becomes

2 x 2 1 2 1+ u′ u′′ (1 + u′ )u′ + (1 + u′ )u =(n 1) (1.1) − 2 2 − u using elementary ODE theory we derive the following identity for any such solution u, graphical over an interval [a, b]

b b 2 ′ 1 1+ u′ (s) R s 1 (1+u 2(z)dz − t 2 u(x) = (n 1) 2 e dsdt (1.2) − x t t u(s) Z Z ′ b R b 1 (1+u 2(z))dz u(b) e− t 2 + x +(u(b) u′(b)b)x dt (1.3) b − t2 Zx Without much diﬃculty, we then show that the quantity (u(b) u′(b)b) is always − positive (which is equivalent to positivity of the mean curvature). This in turn is u(b) equivalent to the monotonicity of the ratios b . Thus, for solutions u on an interval of the form [a, ], we derive the expression ∞

2 ′ ∞ 1 ∞ 1+ u′ (s) R s 1 (1+u 2(z)dz u(x)=(n 1) e− t 2 ds + σx (1.4) − t2 u(s) Zx Zt Existence follows from uniform gradient estimates derived from the formula. Though not eloquent to state, we characterize all graphs over half lines [a, ], and in the ∞ process recover a theorem of Huisken:

Corollary 1 (Huisken’s Theorem from (18)). Let Σn be a smooth self-shrinking hypersurface of revolution, which is generated by rotating an entire graph around the n n 1 n+1 x -axis. Then Σ is the cylinder R S − of radius 2(n 1) in R . 1 × − p 4 Several questions remain open. For example, we believe our methods are sufficient to classify all complete rotationally symmetric embedded self-shrinking surfaces, of which only the sphere and cylinder should be examples. Also, our result motivates study of the Dirichlet problem for the self shrinker equation: Given a curve γ R3, ⊂ when is there a surfaces Σ with ∂Σ= γ satisfying the self-shrinker equation. As an application of our theorem (and in fact the primary motivation for our study of this problem in the first place) we hope to construct self-shrinking solutions in R3 with high genus via gluing constructions developed by N. Kapoleas for de-singularizing intersections of minimal and CMC surfaces. In (26), X. Nguyen constructed an infinite genus surface that translates at constant velocity under mean curvature flow using techniques similar to those to Kapouleas. The analogous construction for self-shrinkers was begun in the articles (27), (28). The construction was never finished, in part because, compared with that in (26), the construction Nguyen attempted in the self-shrinking case lacks a free parameter. The one-parameter family of surfaces that N. Moller and I have discovered give Nguyen’s construction an extra parameter. The general philosophy of a gluing construction is to build an “initial surface” from surfaces you wish to glue together by removing a neighborhood of their intersection and smoothly stitching in a piece of a minimal surface in its place. The resulting initial surface is–depending upon the configuration of the pieces–a good approximation to the geometric problem that you wish to solve. If you wish to construct a new minimal surface in this manner for example then your initial surfaces ought to approximately solve the minimal surface equation. Let us call this initial surface X. We are interested in finding a perturbation X˜ of X that satisfies 1 H ˜ (p)= p ν ˜ (p) X 2 · X ˜ where p is a point in X and HX˜ , νX˜ denote the mean curvature and normal vector to X˜, respectively. In co-dimension one, the problem then reduces to the search for a function φ : X R, giving the surface X˜ as a normal graph of φ over X. φ must → solve the non-linear equation

φ = gij(Dφ(p))D φ(p) p Dφ(p)+ φ(p)=0 Q ij − · 5 in the right function space, which turns out to be a Holder space with symmetries and decay. Unfortunately in general the linearized operator for this equation is not L invertible. In fact, for our initial surface, we expect the linearized operator to be a Fredholm operator of index 1 and trivial kernel. Our initial surface is asymptotically − two conical ends (oriented in three space with axis on the x1-axis) and a plane, with a piece of a singly periodic Scherck surface stitched in near the origin. The extra parameter–that is, the choice of the angle at the vertex of the cone–in our construction can be used to generate and control a function w not contained in the image of L by exploiting the “balancing condition” of (21). Thus, presently, N. Moller and I are working to gratefully extend the work done by X. Nguyen to prove

Conjecture 1.1.4. For a positive integer g suﬃciently large there exists a complete, properly embedded surface without boundary Σg that shrinks by homotheties under motion by mean curvature

In (8), with Maria Calle and Joel Kramer, I considered a class of evolution equations of the form ∂ φ(x, t)= F (λ(x, t))ν(x, t) (1.5) ∂t where φ : M [0, T ) Rn+1 is a one parameter family immersions of a ﬁxed manifold × → M of dimension n into Rn+1 and where the velocity function F satisﬁes a natural set of conditions that, among other things ensure parabolicity of the evolution. In (8), we proved the following,

Theorem 1.1.5. Let Mt t 0 be a one-parameter family of smooth closed and strictly { } ≥ n+1 d convex hypersurfaces in R satisfying dt Mt = F νt, where νt is the unit normal of

Mt and where F satisﬁes conditions stated in (1). Let W (t) denote the width of the

hypersurfaces Mt. Then in the sense of limsup of forward diﬀerence quotients it holds: d C 4π 0 , (1.6) dt ≤ − n and C 4π W (t) W (0) 0 (1.7) ≤ − n

6 This theorem was motivated by the celebrated result of Colding and Minicozzi in (10) concerning the extinction time of finite volume metrics with positive Ricci curvature evolving by Ricci Flow. Similar techniques were used in a follow up paper by Colding and Minicozzi that proved an analogous result for mean curvature flow. The success of their methods towards Ricci and mean curvature flow led naturally to the question of how generally they could be applied. Both Ricci flow and mean curvature flow belong to a wide class of parabolic equations called “geometric” or “curvature flows”. A rather large subclass of these flows (containing mean curvature flow as an example), had been studied by B. Andrews in (1), where Huisken’s famous theorem for mean curvature flow had been generalized; that is, convex surfaces moving by these evolutions collapse in a round point in a finite amount of time.

7 Chapter 2

Background Material

2.1 The Volume Functional, Minimal Surfaces and the Weierstrass Representation

In differential geometry, one of the most natural and often utilized notions is that of the volume of a submanifold, and one of the most natural and studied classes of submanifold are those that, locally at least, minimize volume. More precisely, a submanifold Σ of a manifold M is minimal if its volume is unchanged by infinitesimal deformations with compact support. One sees that if Σ is twice differentiable–a very natural geometric assumption–then Σ is minimal if and only if its mean curvature vector H~ is identically zero. For a parametric surface Φ : M N, the volume Σ → of M is computed by integrating the pullback of the volume form of N by Φ over M M. In local coordinates xi, the induced volume form has the expression dV = i j ∂ ∂ det gijdx dx , where gij = h(dΦ( ),dΦ( )), and h is the metric on N. Let ∧ ∂xi ∂xj denote the space of paremetric surfaces of a fixed dimension (say k) in N. Then X we aim to compute the first variation of the functional Volk : R given by X → k N N Vol ( Φ: M N ) = Φ∗(dV ), where dV is the volume form on N. Thus, { → } M considering a smooth localR perturbation of Φ (that is, a one parameter family of immersions Φ : M Σ N such that the vector field dΦ ( ∂ ) has compact support) t → t ⊂ t ∂t and setting V (t) = Volk( Φ : M N ), we see that the infinitesimal change in { t → }

8 N volume is equal to the time derivative of the volume form Φt∗(dV ) integrated over d d M. In local coordinates, we get dt V (t) = Ω dt det gijdx, where Ω is a coordinate patch. We can assume that at p Ω the metric is the identity matrix at time zero ∈ R (geodesic normal coordinates at p). The cofactor expansion of the determinant gives that

k d d j det gij = ( 1) g1jdet c1j(g) dt t=0 dt − { } j=1 X = g det c (g) + g det c (g) 11,t { 11 } 11 { 11 }t = g + det c (g) 11,t { 11 }t Here, c (g) denotes the (i, j) th cofactor matrix obtained from g. Proceeding induc- ij − d tively, we see that, at p (and at t = 0), we get dt det gij = trace gij,t. Recall that at p, we have normalized the determinant at one; it is easy to check that in general we d have dt det gij = det gijtrace gij,t. We then compute d d ∂ ∂ gij = h dΦ( ),dΦ( ) dt t=0 dt ∂xi ∂xj ∂ ∂ ∂ ∂ = h dΦ( ∂ )dφ( ),dΦ( ) + h dΦ( ∂ )dΦ( ),dΦ( ) ∇ ∂t ∂x ∂x ∇ ∂t ∂x ∂x i j j i ∂ ∂ = 2h dΦ( ), dφ( ∂ )dΦ( ) − ∂t ∇ ∂xi ∂x j Combining this with our previous computations, we see that

d ∂ ∂ det gij = 2h dΦ( ), dφ( ∂ )dΦ( ) det gij (2.1) dt − ∂t ∇ ∂xi ∂x i i ! X where we have used that the coordinate vector ﬁelds ∂ are orthonormal at p ∂xi at time zero, the compatibility of the connection withn theo metric h, and the fact ∇ ∂ the coordinate vector ﬁelds are torsion free. The quantity ∂ dΦ( ) is the i dφ( ) ∂xi ∇ ∂xi mean curvature vector of the immersion at p, and is denotedP by H~ . Thus, we see that an immersion Φ: M N is stationary under normal perturbations for the { → } volume functional Volk if and only if the mean curvature vector vanishes at every d point, in which case we have dt V (t) = 0 for all compactly supported smooth normal

9 perturbations. The mean curvature vector field for any twice differentiable surface is defined at all points on the surface, and is normal to the surface at each point. In an intuitive sense, it measures the net bending of the surfaces in all directions. For example, the mean curvature vector for a sphere is everywhere inward pointing with length equal to the dimension; as expected the mean curvature vector of a plane is everywhere zero, and there exist many non constant saddle-type surfaces that also have everywhere vanishing mean curvature vector, due to competing curvatures pulling in opposite directions. Hence the study of twice differentiable mimimal submanifolds reduces to the the study of the equation H~ = 0. In the case that Σ is a hypersurface, this equation is equivalent to a system of degenerate elliptic PDEs for the coordinate functions of the embedding. For a C2 immersion Φ : M N in which M is an → orientable hypersurface, the unit normal vector field ν Γ(T ⊥M) is well-defined (up ∈ to a sign) and differentiable. In this case, H = H~ ν is called the “mean curvature” · of the immersion. Also, the eigenvalues of the second fundamental form of the immersion are called the “principal curvatures” and are denoted by λ λ , where n 1 ··· n is the dimension of M. Given a domain Ω C, a meromorphic function g on Ω and a holomorphic one- ⊂ form φ on Ω, one obtains a (branched) conformal minimal immersion F : Ω R3, → given by (c.f. (29))

1 1 i 1 F (z)= Re g− (ζ) g(ζ) , g− (ζ)+ g(ζ) , 1 φ(ζ) (2.2) ζ γz ,z 2 − 2 (Z ∈ 0 ) the so-called Weierstrass representation associated to Ω,g,φ. The triple (Ω, g, φ) is

referred to as the Weierstrass data of the immersion F . Here, γz0,z is a path in Ω

connecting z0 and z. By requiring that the domain Ω be simply connected, and that g be a non-vanishing holomorphic function, we can ensure that F (z) does not depend on the choice of path from z to z, and that dF = 0. Changing the base point z has 0 6 0 the eﬀect of translating the immersion by a ﬁxed vector in R3. The unit normal n and the guass curvature of the resulting surface are then (see sections 8, 9 in (29))

10 n = 2Re g, Im g, g 2 1 / g 2 +1 , (2.3) | | − | | 4 ∂ g g K = | z || | . (2.4) − φ (1 + g 2)2 | | | |

Since the pullback F ∗(dx3) is Re φ, φ is usually called the height diﬀerential. The two standard examples are

g(z)= z,φ(z)= dz/z, Ω= C 0 (2.5) \{ } giving a catenoid, and

g(z)= eiz,φ(z)= dz, Ω= C (2.6) giving a heliciod. We will always write our non-vanishing holomorphic function g in the form g = eih, for a potentially vanishing holomorphic function h, and we will always take φ = dz. For such Weierstrass data, the diﬀerential dF may be expressed as

∂xF = (sinh v cos u, sinh vsin u, 1) , (2.7) ∂ F = (cosh v sin u, cosh v cos u, 0) . (2.8) y −

2.2 Mean Curvature Flow

The parabolic analogue of the minimal surface equation, often written schemati- d ~ cally dt x = H(x), is known as mean curvature flow, justified by the fact that solutions to this equation are one-parameter families of immersed submanifolds Σ that evolve { t} in the direction of their mean curvature vector. For our purposes we will assume that the submanifolds are actually hypersurfaces. More explicitly, a family of immersions Φ : M N of a manifold M in to a manifold N (of dimension n) is a mean { t → } ∂ ~ curvature flow if ∂t Φt(p)⊥ = HM (p); that is, the tangential velocity of a point does not play a role. From a geometric point of view, this is a consequence of the fact

11 that the sets Σt := Φt(M) are determined solely by the normal component of vector ∂ ﬁeld V (t, p) := ∂t Φt(p). An easy way to see this is as follows: Take a point (t, p)

where V (t, p)⊥ doesn’t vanish. At such a point, the differential dΦ is non-singular, and hence there is an open set = ( ǫ + t, ǫ + t) U (U M) such that Φ is U − × ⊂ |U a diffeomorphism onto its image φ( ) := Λ. We can also assume that is small U U enough so that its image under Φ is contained in a single coordinate system in N so that we can regard Λ as a subset of euclidean space. We may then define a function F :Λ R by the condition F (q)= τ if and only if q is in the hypersurface Σ . By → τ construction, we have, for p in M, that Φ (p) is in Σ , and thus we get F Φ (p)= t. t t ◦ t Differentiating this equation in t gives that F ∂ Φ (p) = 1. Since, by construction, ∇ · ∂t t the sets Σ Λ are level sets for the function F , we have that F : Rn is non t ∩ ∇ U → vanishing normal vector field at every point; in particular, a quick computation gives that F = 1 H~ . Moreover, if T : Rn is any tangential vector field (that H~ 2 ∇ | | U → is, T (t, p) is in the tangent space TΦt(p)Σt for all t and p), we can evolve points on our initial surface Σ Λ by the vector field V (q) = ∂ Φ (p)+ T (t, p) (where the t ∩ T ∂t t image of the point (t, p) under Φ is q); that is, for q Σ Λ, we can consider the ∈ t ∩ gradient flows q(τ), and qT (τ) for the vector fields V and VT respectively. The above

computation gives that both q(τ) and qT (τ) are in the level set Σt+τ (at least, for small τ, so that the gradient ﬂow is deﬁned). Now at a point interior to the closed

set (t, p) V ⊥(t, p)=0 , the surface Σ is stationary on an open set under the flow, { | } t and hence here we have Σ Λ = Σ for small τ. Thus, the evolutions given by t ∩ τ+t the two vector fields V and VT agree up to level sets and up to a set of measure zero. Continuity then says that the level sets agree wherever they are defined. Mean curvature flow has the natural informal interpretation as the gradient flow for the area functional on submanifolds, i.e., moving a surface in the direction of its mean curvature vector decreases its volume (in the compact case) most rapidly. An easy example of a mean curvature flow that illustrates general phenomena is that of homothetically shrinking concentric spheres, which shrink to a point after a finite amount of time. In general, solutions to mean curvature flow become singular, and it has been a big challenge of the field to understand the nature of the singularities. The regularity theory for mean curvature flow is a vital, intensely researched field that has

12 seen dramatic advances of late. In particular, there has been much recent success in the construction and understanding of various special solutions, so called translating and self-shrinking solutions. Translating and self-shrinking solutions arise as blowup solutions for mean curvature flow. That is, the asymptotic shape of a singularity in mean curvature flow is that of a self shrinking or translating solution. Thus, these special solutions are vital to the understanding the structure of singularities for mean curvature flow.

2.3 Singularities of Mean Curvature Flow and the Equation for Self Shrinkers

As mentioned, most solutions for mean curvature flow Σ become singular in { t} finite time, and thus cease to exist in a classical sense. In (19), Huisken showed that as long as the principal curvatures remain bounded, the flow remains smooth, hence the existence of a singularity implies curvature blowup for the family Σ . { t} Also in (18), it was shown that if the maximum curvature blows up no faster than 1/2 c(T t)− , where T denotes the singular time for the flow, then the asymptotic − shape of the solution will be that of a self-shrinking solution; that is, a sequence of rescaled solutions will converge to hyper-surface satisfying the equation

H = cν X (2.9) · where H denotes the mean curvature vector, ν denotes the unit normal, X denotes the position vector in Rn+1 with respect to an appropriate origin, and c is a positive constant.

2.4 Sweepouts and the notion of Width.

In this section we recall some notions of (9). In (9), Colding and Minicozzi gave a sharp estimate for the extinction time of a convex hypersurface of Rn moving by mean curvature in terms of an invariant called the “width” which we now deﬁne. The following is adapted from (9):

13 Let Φ : M M Rn+1 be a continuous one parameter family of hypersurfaces. t → t ⊂ n 1 1 Take = B − , and define Ω to be the set of continuous maps σ : S M P t ×P → t such that for each s the map σ( ,s) is in W 1,2, such that the map s σ( ,s) is ∈ P · → · continuous from into W 1,2, and finally such that σ( ,s) is a constant map for all s P · ∈ ∂ . We will refer to elements of the set Ω as “sweepouts” of the manifold M . Given P t t a sweepoutσ ˆ Ω representing a non-trivial homotopy class in π M, the homotopy ∈ t n σˆ class Ωt is defined to be the set of all maps σ that are homotopic toσ ˆ through Ωt. We then define the width, W (t), as follows: Fix a sweepout β Ω representing a ∈ t non-trivial homotopy class in M and let β Ω denote the corresponding sweepout t ∈ t of Mt. We then take

W (t)= W (t, β) = inf max Energy (σ( ,s)) (2.10) σ Ωβt s · ∈ t ∈P Here, Energy (σ( ,s)) is the W 1,2-semi norm of the function σ( ,s) given by Energy (σ( ,s)) = · · · d σ(ζ,s) 2dζ. We note that the width is always positive for a manifold with non- S1 | dζ | trivialR fundamental group. In the proof of the main theorem of (8), we will rely heavily on the following theorem of Colding and Minicozzi from (9):

Lemma 2. For each t, there exists a family of sweepouts γj Ω of M that satisfy { } ∈ t t the following:

1. The maximum energy of the slices γj( ,s) converges to W (t). · 2. For each s , the maps γj( ,s) have Lipschitz bound L independent of both j ∈ P · and s.

3. Almost maximal slices are almost geodesics: That is, given ǫ > 0, there exists δ > 0 such that if j > 1/δ and Energy(γj( ,s)) > W (t) δ for some s, then · − there exists a non-constant closed geodesic η in M such that dist(η,γj( ,s)) < ǫ. t · The existence of such a family of sweepouts is established in (9), Theorem 1.9, and we will not include the proof here. The distance in the third statement of the previous Lemma is given by the W 1,2 norm on the space of maps from S1 to M.

14 Chapter 3

Width and Mean Curvature Flow

It is well known that a convex closed hypersurface in Rn+1 evolving by its mean curvature remains convex and vanishes in ﬁnite time. In (1), Ben Andrews showed that the same holds for much broader class of evolutions. T. H. Colding and W. P. Minicozzi in (9) give a bound on extinction time for mean curvature ﬂow in terms of an invariant of the initial hypersurface that they call the width. In this paper, we generalize this estimate to the class evolutions considered by Andrews:

Theorem 3. Let Mt t 0 be a one-parameter family of smooth compact and strictly { } ≥ n+1 d convex hypersurfaces in R satisfying dt Mt = F νt, where νt is the unit normal of

Mt and where F satisﬁes conditions 3.1 in (1). Let W (t) denote the width of the

hypersurface Mt. Then in the sense of limsup of forward diﬀerence quotients it holds: d C 4π W 0 , (3.1) dt ≤ − n and C 4π W (t) W (0) 0 t. (3.2) ≤ − n All functions F in this class are assumed to be either concave or convex. In the concave case, we require an a priori pinching condition on the initial hypersurface

M0. In the case of convexity a pinching condition is not needed, and we can take

C0 = 1 above. This class is of interest since it contains many classical ﬂows as particular examples, such as the n-th root of the Gauss curvature, mean curvature, and

15 hyperbolic mean curvature flows. The key to arriving at our estimate on extinction time is a uniform estimate on the rate of change of the width, for which preservation of convexity of the evolving hypersurface is fundamental. We also study flows which are similar to those considered by Andrews in (1), except that we allow for higher degrees of homogeneity. The motivation for this consideration is that the degree 1 homogeneity condition on the flows given in Andrews’ paper excludes some naturally arising evolutions, such as powers of mean curvature, for which preservation convexity and extinction time estimates are known (see (30)). An analogous result as (3.1) holds for the time zero derivative of width of an initially convex hypersurface for these kinds of flows. However in the general case it is, to our knowledge, unknown as to whether convexity is preserved, or even if there is a well-defined extinction time. Consequently, nothing can be said about the long term behavior of the width of a hypersurface evolving under these flows. We conclude by taking the 2-width to be, loosely speaking, the area of the smallest homotopy 2-sphere required to pull over the surface, and we show that an analogous estimate on the derivative of the 2-width holds, and that in fact in this case the convexity assumption can be loosened to 2-convexity.

3.1 Contraction of Convex Hypersurfaces in Rn+1

Let M n be a smooth, closed manifold of dimension n 2. Given a smooth ≥ immersion φ : M n Rn+1, we consider a smooth family of immersions φ : M n 0 → × [0, T ) Rn+1 satisfying an equation of the following form: →

∂ φ(x, t) = F (λ(x, t))ν(x, t) (3.3) ∂t φ(x, 0) = φ0(x)

for all x M n and t [0, T ). In this equation ν(x, t)= H / H is the inward ∈ ∈ Mt | Mt | n n pointing unit normal to the hypersurface Mt := φ(M , t) = φt(M ) at the point

φt(x), and λ(x, t)=(λ1(x, t), ..., λn(x, t)) are the principal curvatures of Mt at φt(x), that is, the eigenvalues of the Weingarten map of the immersion at that point.

16 We consider velocity functions F satisfying the following hypotheses, taken from (1):

Conditions 1. 1. F : Γ R is deﬁned on the positive cone Γ = λ = + → + { (λ , ..., λ ) Rn : λ > 0, i =1 ...n . 1 n ∈ i } 2. F is a smooth symmetric function.

3. F is strictly increasing in each argument: ∂F > 0 for i =1 ...n at every point ∂λi

in Γ+.

4. F is homogeneous of degree one: F (cλ)= cF (λ) for any positive c R. ∈

5. F is strictly positive on Γ+, and F (1,..., 1)=1.

6. one of the following holds:

(a) F is convex; or

(b) F is concave and either

i. n =2;

ii. f approaches zero on the boundary of Γ+; or

H Σλi iii. supx M F < lim infλ ∂Γ+ F (λ) ∈ ∈ Given an initially convex smooth immersion φ : M n Rn+1 and a function F 0 → satisfying conditions 1 , B. Andrews proved (Corollary 3.6, Theorem 4.1 and Theorem 6.2 in (1)) that there exists a unique family of smooth immersions φ(x, t): M n × [0, T ) Rn+1 satisfying equations (3.3) for all x M n. Moreover, he proved that → ∈ the embeddings M remain convex and converge uniformly to a point as t T . t →

3.2 Width and the existence of Good Sweepouts

In this section we recall some notions of (9). Convexity of the immersion φ : M M Rn+1 implies that M is diﬀeomorphic t → t ⊂ n n 1 n 1 to the n-sphere S , via the Gauss map. S is then equivalent to S B − / , where × ∼

17 1 n 1 is the equivalence relation (θ ,y) (θ ,y) where θ , θ S and y ∂B − . Here ∼ 1 ∼ 2 1 2 ∈ ∈ n 1 n 1 n B − is the unit ball in R − . We use this decomposition of M = S to define the width W (t) of the immersion φt. n 1 1 Take = B − , and define Ω to be the set of continuous maps σ : S M P t ×P → t such that for each s the map σ( ,s) is in W 1,2, such that the map s σ( ,s) is ∈ P · → · continuous from into W 1,2, and finally such that σ( ,s) is a constant map for all s P · ∈ ∂ . We will refer to elements of the set Ω as “sweepouts” of the manifold M . Given P t t a sweepoutσ ˆ Ω representing a non-trivial homotopy class in π M, the homotopy ∈ t n σˆ class Ωt is defined to be the set of all maps σ that are homotopic toσ ˆ through Ωt. We then define the width, W (t), as follows: Fix a sweepout β Ω representing a ∈ t non-trivial homotopy class in M and let β Ω denote the corresponding sweepout t ∈ t of Mt. We then take

W (t)= W (t, β) = inf max Energy(σ( ,s)) (3.4) σ Ωβt s · ∈ t ∈P We note that the width is always positive until extinction time. In the proof of our main theorem, we will rely heavily on the following

Lemma 4. For each t, there exists a family of sweepouts γj Ω of M that satisfy { } ∈ t t the following:

1. The maximum energy of the slices γj( ,s) converges to W (t). · 2. For each s , the maps γj( ,s) have Lipschitz bound L independent of both j ∈ P · and s.

18 3.3 F convex

We assume the velocity function F satisﬁes conditions (1) through (5) above, in addition to condition (6a). We then have the following:

Lemma 5. For all (λ , ..., λ ) Γ , it holds 1 n ∈ + n λ F (λ , ..., λ ) 1 i . 1 n ≥ n P

Proof. Let λ = (λ1, ..., λn). Let Sn the group of permutations of (1, ..., n), and let λ =(λ , ..., λ ) for each σ S . Then, since F is symmetric, F (λ )= F (λ) for σ σ(1) σ(n) ∈ n σ all σ Sn. ∈ n Let h = P1 λi , let h˜ =(h, ..., h) Γ . The we have the following: n ∈ +

λσ = ( λσ(1), ..., λσ(n))

σ Sn σ Sn σ Sn ∈ ∈ ∈ X X n Xn

= Sn 1 ( λi, ..., λi)= n Sn 1 h˜ = Sn h,˜ (3.5) | − | | − | | | 1 1 X X where the last inequality follows from Sn = n! = n(n 1)! = n Sn 1 . Then, since | | − | − | 1 < 1, by convexity we have: Sn | |

1 Sn F (h˜) F (λσ)= | |F (λ)= F (λ). (3.6) ≤ Sn Sn σ Sn | | X∈ | | But then, by the forth and ﬁfth property of F we have that h = hF (1, ..., 1) = F (h˜), and therefore we obtain the result.

We can then prove the following estimate (corollary 2.9 in (9)):

Lemma 6. Let Σ M be a closed geodesic and Σ M the corresponding evolving ⊂ 0 t ⊂ t closed curve. Let Et be the energy of Σt. Then: d 4π Et . (3.7) dt t=0 ≤ − n

19 Proof. Observe that, since M0 is convex and Σ is minimal, HΣ points in the same direction as ν. By convexity, we have that H H . Let V be the length of the | Σ|≤| M0 | t closed curve Σt. Then, by the first variation formula for volume (see 9.3 and 7.5’ in (31)) we have the following: d Vt = HΣ, F (x, t)ν(x, t) dt t=0 − h i ZΣ 1 1 H H H 2. (3.8) ≤ −n | Σ|| M0 | ≤ −n | Σ| ZΣ ZΣ Here the first inequality follows from Lemma 5, and the second inequality follows from 0 H H . Then we can compute the variation of energy as follows: ≤| Σ|≤| M0 | 2 2 d d V0 2 1 4π π Et = V0 Vt HΣ HΣ . (3.9) dt t=0 dt t=0 ≤ − n | | ≤ −n | | ≤ − n ZΣ ZΣ Here the first inequality follows from (3.8), the second from the Cauchy-Schwarz inequality, and the last inequality follows since by Borsuk-Fenchel’s theorem every closed curve in Rn+1 has total curvature at least 2π (see (7), (16)).

3.4 F concave

We assume the velocity function F satisﬁes conditions (1) through (5) above, in addition to condition (6b). We also require that the initial convex immersion φ : M Rn+1 satisﬁes an a priori pinching condition 0 →

max λ (x, 0), ..., λ (x, 0) C min λ (x, 0), ..., λ (x, 0) (3.10) { 1 n } ≤ 0 { 1 n } for all x M. Here we have chosen an orientation on M so that the sign of the ∈ principle curvatures is positive. We then have

HM0 sup | | = C0. (3.11) x M0 nF ∈ We then have the following:

Lemma 7. For all t 0, ≥ HMt HM0 sup | | sup | | =: C0. x Mt nF ≤ x M0 nF ∈ ∈

20 Proof. By the parabolic maximum principle. See proof of Theorem 4.1 in (1).

We can then prove the following estimate:

Lemma 8. Let Σ M be a closed geodesic and Σ M the corresponding evolving ⊂ 0 t ⊂ t closed curve. Let Et be the energy of Σt. Then: d 4π Et . (3.12) dt t=0 ≤ −nC0 Proof. The proof follows exactly as in the proof of Lemma 6, except that here Lemma 7 gives the inequality d 1 2 Vt HΣ (3.13) dt t=0 ≤ −nC | | 0 ZΣ We then have as before

2 2 d d V0 2 1 4π π Et = V0 Vt HΣ HΣ . (3.14) dt t=0 dt t=0 ≤ −nC | | ≤ −nC | | ≤ −nC 0 ZΣ 0 ZΣ 0 3.5 Extinction Time

We can now prove our main theorem:

Theorem 9. Let Mt t 0 be a one-parameter family of smooth compact and strictly { } ≥ n+1 convex hypersurfaces in R ﬂowing by equation (3.3). Let C0 = 1 if F is convex,

and C0 as in Lemma 7 if F is concave. Then in the sense of limsup of forward diﬀerence quotients it holds: d 4π W , (3.15) dt ≤ −nC0 and 4π W (t) W (0) t. (3.16) ≤ − nC0 As a consequence, we get the following bound on the extinction time:

Corollary 10. Let Mt t 0 be as above, then it becomes extinct after time at most { } ≥ nC W (0) 0 . 4π

21 Observe that in the concave case, the constant C0 depends on the pinching of the

initial hypersurface M0, whereas in the convex case the bound on the extinction time is independent of the evolving hypersurface. We will need the following consequence of the ﬁrst variation formula for energy in 2 the proof of our main theorem: If σt, ηt are two families of curves evolving by a C vector ﬁeld V, then

d d 2 Energy(η ) Energy(σ ) C V 2 σ η 1,2 (1 + sup σ′ ) (3.17) |dt t − dt t | ≤ || ||C || t − t||W | t| Proof of Theorem 9. Here we follow the outline of Theorem 2.2 in (9) Fix a time τ. Throughout the proof, C will denote a constant depending only on j Mτ , but will be allowed to change from inequality to inequality. Let γ be a sequence of sweepouts in M given by Lemma 4. For t τ, let σj(t) denote the curve in τ ≥ s M corresponding to γj = γj( ,s), and set e (t) = Energy(σj(t)). We will use the t s · s,j s following claim to establish an upper bound for the width: Given ǫ > 0, there exist δ > 0 and h > 0 so that if j > 1/δ and 0

4π 2 es,j(τ + h) max es0,j(τ) − + Cǫ h + Ch . (3.18) − s0 ≤ nC 0 The result follows from (3.18) as follows: take the limit as j in (3.18) to get →∞ W (τ + h) W (τ) 4π − − + Cǫ + Ch. (3.19) h ≤ nC0 Taking ǫ 0 in (3.19) gives (3.15). → It remains to establish (3.18). First, let δ > 0 be given by Lemma 4. Since β 2 is homotopically non-trivial in Mτ , W (τ) is positive and we can assume that ǫ < W (τ)/3 and δ < W (τ)/3. If j > 1/δ, and e (τ) > W (τ) δ, then Lemma 4 gives s,j − j a non-constant closed geodesic η in Mτ with dist(η,γs) < ǫ. Letting ηt denote the

image of η in Mt, we have, using (3.17) with V = F ν and the uniform Lipschitz bound L for the sweepouts at time τ, that

d d 2 4π es,j(t) Energy(ηt)+ Cǫ F ν C2 (1 + L ) − + Cǫ (3.20) dt t=τ ≤ dt t=τ || || ≤ nC0 j j j Since σs(t) is the compositon of γs with the smooth ﬂow and γs has Lipschitz bound

L independent of j and s, the energy function es,j(τ + h) is a smooth function of h

22 with a uniform C2 bound independent of both j and s near h = 0. In particular, the

Taylor expansion for es,j(τ + h) gives d e (τ + h) e (τ)= e (τ)h + R (e (τ + h)) (3.21) s,j − s,j dh s,j 1 s,j

2 where R1(es,j(τ + h)) denotes the ﬁrst order remainder. The uniform C bounds on es,j(t) give uniform bounds on the remainder terms R1(es,j(τ + h)), and using (3.20), we get 4π e (τ + h) e (τ) − + Cǫ h + Ch2 (3.22) s,j − s,j ≤ nC 0 from which the claim follows. In the case e (τ) W (τ) δ, the claim (3.18) s,j ≤ − automatically holds after possibly shrinking h0 > 0:

es,j(τ + h) max es0,j(τ) es,j(τ + h) es,j(τ) δ (3.23) − s0 ≤ − − 4π Taking h sufficiently small, so that δ − h, and using the differentiability of − ≤ nC0 e (t), we get (3.18). To get (3.16), note that for any ǫ > 0, the set t W (t) s,j { | ≤ W (0) ( 4π ǫ)t contains 0, is closed, and (3.15) implies that is also open: Take − nC0 − } F (t)= W (t) W (0)+( 4π ǫ)t, and note that d F (t) < 0 in the sense of limsup of − nC0 − dt forward difference quotients. We therefore have W (t) W (0) ( 4π ǫ)t for all t ≤ − nC0 − up to extinction time. Taking ǫ 0 gives (3.16). →

3.6 F homogeneous of degree k

We assume the function F has the same properties as in section 3.3, except that it is homogeneous of degree k for some integer k > 0, i.e., F (cλ1, ..., cλn) = k c F (λ1, ..., λn). Then we can prove the following lemma:

Lemma 11. For all (λ , ..., λ ) Γ , it holds 1 n ∈ + n λ k F (λ , ..., λ ) 1 i . 1 n ≥ n P Proof. Analogous to the proof of Lemma 5.

23 We can then prove the following estimate (analogous to lemma 6):

Lemma 12. Let Σ M be a closed geodesic and Σ M the corresponding evolving ⊂ 0 t ⊂ t closed curve. Let Et be the energy of Σt. Then:

k+1 k+1 d 2 k +1 Et (2π) 2 . (3.24) dt t=0 ≤ − nk

Proof. Observe that, since M0 is convex and Σ is minimal, the mean curvature vector of Σ in Rn+1 points in the same direction as ν. We have that H H by | Σ|≤| M0 | convexity. Let Vt be the length of the close curves Σt, we can use the ﬁrst variation formula for volume (see 9.3 and 7.5’ in (31)) to obtain the following:

1 d k+1 k 1 k k Vt = V0 HΣ, F (x, t)ν(x, t) V0 HΣ HM0 (3.25) k +1 dt t=0 − h i ≤ −nk | || | ZΣ ZΣ 1 1 k+1 1 V k H k+1 H (2π)k+1. ≤ −nk 0 | Σ| ≤ −nk | Σ| ≤ −nk ZΣ ZΣ Here the ﬁrst inequality follows from Lemma 5, the second inequality follows from 0 ≤ H H , the third one follows from H¨older’s inequality and the last inequality | Σ|≤| M0 | follows as above from Borsuk-Fenchel’s theorem. Then we can compute the variation of energy as follows:

d k+1 1 d k +1 k+1 2 k+1 2 Et = k+1 Vt k (2π) . (3.26) dt t=0 (2π) 2 dt t=0 ≤ − n

Finally, we could use this estimate to give a bound on the decrease rate of the width as in Theorem 9, to obtain the following:

Theorem 13. Let Mt t 0 be a one-parameter family of smooth compact and strictly { } ≥ convex hypersurfaces in Rn+1 ﬂowing by equation (3.3), with F as in this section. Then in the sense of limsup of forward diﬀerence quotients it holds:

d k+1 k +1 k+1 W (2π) 2 , (3.27) dt ≤ − nk and k +1 k+1 W (t) W (0) (2π) 2 t (3.28) ≤ − nk for as long as the evolving manifold remains smooth and strictly convex.

24 3.7 2-Width and 2-Convex Manifolds

Until now, we have assumed that the evolving manifold is strictly convex. Also, we have defined the width using sweepouts by curves. We can generalize the result for manifolds which are “2-convex”, that is, such that the sum of any two principal curvatures is positive, by using a different width, defined by sweepouts of 2-spheres. The fundamental point is that in the case of 2-width, as earlier, one can rely on the existence of a sequence of “good sweepouts” as comparisons in order to estimate the derivative of the width. For higher dimensional sweepouts the analogous result is not known, and so we cannot argue as before in trying to prove an extinction time estimate. A motivation for dealing with general higher dimensional widths, as opposed to 1-width, is that it allows for a relaxation of the convexity condition. In defining the width above, we chose to decompose Sn topologically into the space 1 n 1 1 n 1 S B − , with each set (t, s): t S collapsed to a point, for each s ∂B − . × { ∈ } ∈ The purpose of this decomposition is that, since M is a closed convex hypersurface, it is topologically Sn, and the decomposition ensures that the sweepouts induce non- n trivial maps on πn(S ) - i.e. they are not homotopically trivial - and hence that the width is positive until the hypersurface becomes extinct. If we require that the initial immersion φ : M Rn+1 is only 2-convex, we lose information about the global 0 → topology of M, and so it makes sense to use more general parameter spaces than those that ensure the sweepouts are homotopy Sn’s. Let be a compact finite dimensional topological space, and let Ω be the set P t of continuous maps σ : S2 M so that for each s the map σ( ,s) is in ×P → ∈ P · C0 W 1,2(S2, M), the map s σ( ,s) is continuous from to C0 W 1,2(S2, M), and ∩ → · P ∩ finally σ maps ∂ to point maps. Given a mapσ ˆ Ω , the homotopy class Ωσˆ Ω P ∈ t t ⊂ t is defined to be the set of maps σ Ω that are homotopic toσ ˆ through maps in ∈ t Ω . To define the 2-width, fix a homotopy class β Ω representing a non-trivial t ∈ 0 homotopy class in M and let β Ω represent the corresponding sweepout in M . 0 t ∈ t t Then, for each t, we define

W2(t)= W2(t, σˆ) = inf max Energy(σ( ,s)), (3.29) σ Ωσˆ s · ∈ t ∈P

25 where the energy is given by

1 2 Energy(σ( ,s)) = xσ(x, s) dx. (3.30) · 2 S2 |∇ | Z It was shown in (10) and (11) that we could also deﬁne the energy using area, and we would obtain the same quantity:

A W2 = inf max Area(σ( ,s)) = W2. (3.31) σ Ωσˆ s · ∈ t ∈P As in the case of 1-width deﬁned above, we will use the existence of a sequence of “good sweepouts” that approximate the width. Namely, the following theorem was proven in (10), Theorem 1.14:

Theorem 14. Given a map β Ω representing a non-trivial class in π (M ), there ∈ t n t j β j exists a sequence of sweepouts γ Ωt with maxs Energy(γ ( ,s)) W2(β), and ∈ ∈P · → so that given ǫ> 0, there exist ¯j and δ > 0 so that if j > ¯j and

Area(γj( ,s)) > W (β) δ, (3.32) · 2 − then there are ﬁnitely many harmonic maps u : S2 M with i → t d (γj( ,s), u ) < ǫ. (3.33) V · ∪i{ i}

Here dV denotes varifold distance as deﬁned in (10).

We consider now a family M n Rn+1 evolving by the evolution equation (3.3). t ∈ However, now the manifolds Mt are not necessarily convex, but they are 2-convex: we may choose an orientation on the ambient space Rn+1 and on M so that the sum of any two principal curvatures is always positive. Equivalently, if the principal curvatures are λ λ ... λ , then λ λ . Observe that this implies that 1 ≤ 2 ≤ ≤ n | 1| ≤ 2 Mt is mean convex, that is, the mean curvature is positive in this orientation. We assume also that n> 3. The velocity function F now is deﬁned in all Rn, and satisﬁes the rest of conditions in Section 3.1. Observe that the third condition on F ensures that the evolution equation is parabolic, and therefore a smooth solution exists at least on a short time interval (see Theorem 3.1 in (20)). Also, Lemma 5 or 7 still hold in this case. Then we can generalize Lemmas 6 and 8 as follows:

26 Lemma 15. Let Σ M be a closed minimal surface and Σ the corresponding ⊂ 0 t surface in Mt. Let Vt = Area(Σt). Then: d 16π Vt , (3.34) dt t=0 ≤ −nC0 where C0 is 1 if F is convex, and it is as in Lemma 7 if F is concave.

Proof. Observe that, since Σ is minimal in M0, the mean curvature vector of Σ is

perpendicular to M0. Let k1 and k2 be the principal curvatures of Σ in M0, that is,

HΣ = k1 + k2. Then by 2-convexity (and because n> 3), we can chose an orientation on Rn+1 and M so that: n

0 k1 + k2 λn 1 + λn (λ1 + λ2)+ λn 1 + λn λi. (3.35) ≤ ≤ − ≤ − ≤ 1 X This shows that H and H in fact point in the same direction, with 0 H Σ M0 ≤ | Σ| ≤ H . As before, we can then compute the ﬁrst variation of volume as: | M0 | d 1 Vt = HΣ, F (x, t)ν(x, t) HΣ HM0 (3.36) dt t=0 − h i ≤ −n | || | ZΣ ZΣ 1 16π H 2 , ≤ −n | Σ| ≤ − n ZΣ where the last inequality is from, e.g. (32).

Then we can prove the following theorem:

Theorem 16. Let Mt t 0 be a one-parameter family of smooth compact 2-convex { } ≥ hypersurfaces in Rn+1 (with n > 3) ﬂowing as represented in equation (3.3). Then in the sense of limsup of forward diﬀerence quotients it holds: d 16π W (3.37) dt 2 ≤ − n and 16π W (t) W (0) t (3.38) 2 ≤ 2 − n for as long as the solution remains smooth and 2-convex.

This result gives a bound for the time of the ﬁrst singularity. Unlike in the convex case, we don’t know that the submanifold contracts into a point by the ﬁrst singularity. The proof is analogous to the proof of Theorem 9 (see also (10) and (11)).

27 Chapter 4

A Singular Lamination

In this chapter we record the alternate proof, recorded in (24), of the main theorem of D. Hoﬀman and B. White in (17). The theorem generalizes an example of Colding and Minicozzi in (12) that was the ﬁrst known example of a sequence of properly embedded minimal surfaces converging to a non-proper limit:

Theorem 17. Let M be a compact subset of x = x = 0, x < 1/2 and let { 1 2 | 3| } C = x2 +x2 =1, x < 1/2 . Then there is a sequence of compact embedded minimal { 1 2 | 3| } disks Σ C with ∂Σ ∂C and containing the vertical segment (0, 0, t) t < 1/2 k ⊂ k ⊂ { || | } so that:

It follows from (D) that the Σ M converge to a limit lamination of C M. The k \ \ leaves of this lamination are given by the multi-valued graphs ΣI given in (D), indexed

28 by intervals I of the compliment of M, taken together with the planes x = t C { 3 } ∩ for t M. This lamination does not extend to a lamination of C, however, as every ∈ boundary point of M constitutes a non-removable singularity. Theorem 17 is inspired by the result of Hoﬀman and White in (17). 3 Throughout this chapter, we will use coordinates (x1, x2, x3) for vectors in R , and z = x + iy on C. For p R3, and s > 0, the ball in R3 is B (p). We denote ∈ s 3 the sectional curvature of a smooth surface Σ by KΣ. When Σ is immersed in R ,

AΣ will be its second fundamental form. In particular, for Σ minimal we have that A 2 = 2K . Also, we will identify the set M x -axis with the corresponding | Σ| − Σ ⊂{ 3 } subset of R C; that is, the notation will not reﬂect the distinction, but will be ⊂ clear from context. Our example will rely heavily on the Weierstrass Representation, which we introduce here.

4.1 The Structure of the Proof

We will be dealing with a family of immersions F : Ω R3 that depend k,a k,a → on a parameter 0 0 when y =0. A k,a k,a k,a 6 look at the expression for the unit normal given above in (2.3) then shows that all of the surfaces Σ will be multi-valued graphs away from the x -axis (since g(x, y) =1 k,a 3 | | is equivalent to y = 0). The dependence on the parameter 0

29 Lemma 18. (a) The horizontal slice x = t F (Ω ) is the image of the vertical { 3 } ∩ k k segment x = t in the plane, i.e., x (F (t, y)) = t. { } 3 k (b) The image of F ( x = t ) is a graph over a line segment in the plane x = k { } { 3 t (the line segment will depend on t) }

This gives that the immersions F : Ω R3 are actually embeddings, and k k →

that the surfaces Σk given by Fk(Ωk) are all embedded in a ﬁxed cylinder Cr0 = x2 + x2 = r2, x < 1/2 about the x -axis in R3. This will then imply that the { 1 2 0 | 3| } 3 surfaces Σ converge smoothly on compact subsets C M to a limit lamination of k r0 \

Cr0 . The claimed structure of the limit lamination(that is, that on each interval of the compliment it consists of two multi-valued graphs that spiral into planes from above and below) will be established at the end.

Figure 4.1 The functions Fk

Throughout the paper, all computations will be carried out and recorded only on the upper half plane in C, as the corresponding computations on the lower half plane are completely analogous. By scaling it suﬃces to prove Theorem 17 (D) for some

Cr0 , not C1 in particular.

30 4.2 deﬁnitions

Let M [0, 1] be a closed set. Fix γ > 1, and take M 1 to be the empty set. ⊂ − Then for k a non-negative integer, we inductively deﬁne two families of sets mk, and

Mk as follows: Assuming mk 1 and Mk 1 are already defined, take mk to be any − − maximal subset of M with the property that, for p, q mk, r Mk 1, it holds that ∈ ∈ − k p q , p r γ− . Then define Mk = Mk 1 mk and M = kMk. Take | − | | − | ≥ − ∪ ∞ ∪ z dz h(z)= = u(z)+ iv(z) (4.1) (z2 + a2)2 Z0 and 2 2 5/4 y0(x)= ǫ x + a for ǫ to be determined. For p R we define ∈ h (z)= h(z p)= u (z)+ iv (z) p − p p and y (x)= y (x p) p 0 − We then take

hl(z)= hp(z)= ul(z)+ ivl(z) (4.2) p m X∈ l and

yl(x)= minp m yp(x) ∈ l We take k l Hk(z)= µ− hl(z)= Uk(z)+ iVk(z) (4.3) Xl=0 for a parameter µ>γ to be determined. We take

Yk(x) = minl kyl(x) ≤

Note that the dependence on the parameter a is ommited from the notation. We take ω = x + iy y y (x) ,ω = x + iy y y (x) { || | ≤ 0 } p { || | ≤ p }

31 l Figure 4.2 A schematic rendering of the domain Ω in the case of M = p = 2− l k { l − | ∈ N } and ω = x + iy y y (x) , Ω = x + iy y Y (x) l { || | ≤ l } k { || | ≤ k } and lastly set Ω = kΩk. ∞ ∩

4.3 Preliminary Results

We record some elementary properties of the sets Mk and mk deﬁned above which will be needed later.

Lemma 19. m γk +1 | k| ≤

Proof. Let p1 <...

n 1 − k 1 p p = p p (n 1)γ− ≥ n − 1 k+1 − k ≥ − Xk=1

k Lemma 20. For all p in M, there is a q in M such that p q <γ− k | − |

Proof. If not, mk is not maximal.

32 Lemma 21. The union ∞ m = ∞ M M ∞ is a dense subset of M ∪k=0 k ∪k=0 k ≡ Proof. Suppose not. Then there is a q M, and a positive integer k such that ∈ k p q >γ− , p M ∞. In particular, this implies that m is not maximal. | − | ∀ ∈ k In order to avoid disrupting the narrative, the proofs of the remaining results in this section will be recorded later in the Appendix at the end. The proofs are somewhat tedious, though easily veriﬁed.

Lemma 22. For ǫ suﬃciently small, hp(z) is holomorphic on ωp, hl is holomorphic

on ωl, and Hk is holomorphic on Ωk. We will also need the following estimates:

Lemma 23. On the domain ωp it holds that: ∂ c x p y u (x, y) 1| − || | ∂y p ≤ ((x p)2 + a2)3

− and ∂ c v (x, y) > 2 ∂y p ((x p)2 + a2)2 − Integrating the above estimates from 0 to the upper boundary of ωp gives

u (x, y (x)) u (x, 0) ǫ2c p p − p ≤ 1 and ǫc min v(x, y) > 2 [yp(x)/2,yp(x)] ((x p)2 + a2)3/4 − These estimates immediately give

Corollary 24. k 2 l l 2 U (x, Y (x)) U (x, 0) ǫ c (γ/µ) + µ− ǫ c′ (4.4) | k k − k | ≤ 1 ≤ 1 Xl=0

k ǫc2 l Yk(x) 2 2 3/4 1 Vk(x, Yk(x)/2) > µ− ((x p) + a )− c2qk(x) (4.5) 2 yp(x) − ≡ 2 l=0 p m X X∈ l Here, the quantity qk is deﬁned by the last equality. Lastly we record Lemma 25. Yk(x) qk(x) qk 1(x) (4.6) ≥ Yk 1(x) − −

33 4.4 Proof of lemma 18

We will ﬁrst concern ourselves with establishing lemma 18. (a) follows from (2.2) 1/2 and the choice of z0 = 0 Choosing ǫ < ǫ0 < c1′− , where c1′ is the constant in (4.4), and using (2.8) we get

∂ F (x, y), ∂ F (x, 0) = cosh V (x, y) cos(U (x, y (x) U (x, 0)) > cosh V (x, y)/2 h y k y k i k k 0 − k k (4.7) Here we have used that cos(1) > 1/2. This gives that all of the maps F : Ω R3 k k → are indeed embeddings (for all values of a) and proves (b) of lemma 18.

Now, integrating (4.7) from Yk(x)/2 to Yk(x) gives

Yk(x) min V F (x, Y (x)) F (x, 0), ∂ F (x, 0) > e [Yk/2,Yk] k (4.8) h k k − k y k i 2

Using the bound for Vk recorded in (4.5), we get that

Yk(x) 1 c q (x) F (x, Y (x)) F (x, 0), ∂ F (x, 0) > e 2 2 k (4.9) h k k − k y k i 2

1 1 c q (x) Take r (x) Y (x)e 2 2 k . We will show that, for an appropriate sequence a 0, k ≡ 2 k k → rk(x) ckrk 1(x) for a sequence ck with 0 < ck < 1 such that ∞ cl > c0; this ≥ − l=0 establishes (c) of lemma 18. Q

There are two cases to consider when comparing rk(x) with rk 1(x). If Yk 1(x)= − − Yk(x), there is nothing to do. If Yk(x) < Yk 1(x) then we may assume that Yk(x) = − yp(x) for some p mk. This is the case we are concerned with. ∈ −k iµ hp(z) Consider the immersion given by data g(z) = e ,φ = dz, with domain ωp. Arguing as above, we can estimate radius of the cylinder in which it is embedded by

−k 1 2 2 −3/4 1 2 2 5/4 ǫc2µ ((x p) +a ) r −k (x)= ǫ((x p) + a ) e 2 − p,µ 2 − which we rewrite as

1 −k −1 ǫ 5/3 ǫc2µ s r −k (s(x)) = s e 2 (4.10) p,µ 2 2 2 3/4 k with s = ((x p) + a ) . The right hand side of (4.10) has a minimum at s = cµ− − 5/3k with value cµ− (here c is used to denote potentially diﬀerent positive constants

34 and should be thought of as denoting “on the order of”). as k goes to inﬁnity, the embedded surfaces Σp,µ−k ”pinch oﬀ” at the singular point, and the limiting immersion is not embedded in any neighborhood of our line segment. On the other hand, to complete our construction, we are forced “scale” the data by uniformly summable factors, so that the curvature remains bounded away from the singular set. Lemmas 26 and 27 show that we can choose a σ so that as long as x p < | − | 2/3(1+σ)k cµ− (with a small enough, σ > 0), then rp,µ−k (x) > 1. In the following we set ǫ −1 ǫ 5/3 2 c2s Φ(s)= 2 s e .

Lemma 26. α> 0, δ = δ(α) s.t. ∀ ∃ α Φ(s) s− (4.11) ≥ for 0

Proof. 1 −1 ǫ 5/3+α ǫc2s lims 0 s e 2 = → 2 ∞ for all α.

The point now is to choose µ and σ so that µ2/3 < µ(1+σ)2/3 <γ<µ. We must also choose α so that ασ 5/3 0, as will be seen in the following. In fact, for later − ≥ applications, we demand ασ 5/3 1. − ≥ Lemma 27. For 2/3 2/3(1+σ)k δ(α) x p , a µ− | − | ≤ √ 2 We have that

rp,µ−k (x) > 1

Proof. The assumptions immediately give that

2 2 3/4 (1+σ)k s = ((x p) + a ) <µ− δ < δ − which we re-write as

k σk µ s µ− δ ≤ 35 applying (4.11), we ﬁnd that

k σk α Φ(µ s) > µ− δ −

Equivalently,

ǫ −k ǫ 5/3 c2µ s 1 (ασ 5/3)k α r −k (x)= s e 2 − >µ − δ− > 1 p,µ 2 since we have choosen ασ 5/3 1, and we may assume δ < 1. − ≥ We are ready to prove:

µ(1+σ)2/3 2/3 √ Lemma 28. set τ = γ < 1 and c0 = δ / 2. Then either

cτ k 1 5/2 ǫc5/2µ 5/3k r (x) 0 − k 1 k ≥ 1+ γc− τ 2 0 " # or cτ k 1 5/2 ǫc5/2µ 5/3k r (x) > 0 − r (x) k 1 k k 1 1+ γc− τ 2 − 0 " # where

2/3(1+σ)k 2/3 Proof. We may assume Y (x)= y (x) for some p m . If x p ,a<µ− δ /√2, k p ∈ k | − | then

rk(x) >rp,µ−k (x) > 1

2/3(1+σ)k 2/3 by lemma 27. So we assume that x p > c µ− with c = δ /√2. By the | − | 0 0 k+1 construction of the sets mk, Mk, there is a q Mk 1 such that p q < γ− . We ∈ − | − | may assume that q is the closest point to x in Mk 1 so that Yk 1(x)= yq(x). − − Then we may estimate

4/5 4/5 2 2 Yk(x) yp(x) x p + a = > | − | 2 2 Yk 1(x) yq(x) ( x p + p q ) + a − | − | | − | or

4/5 2 2 Yk(x) 1+ a / x p 1 > | −2 | 2 2 > (4.12) Yk 1(x) (1 + p q / x p ) + a / x p (1 + p q / x p ) − | − | | − | | − | | − | | − | 36 Figure 4.3 The “good interval about p”

The last inequality in (4.12) follows since

d 1+ x > 0 dx c + x k+1 2/3(1+σ)k for c > 1 and x > 0. Recalling that p q < γ− , x p > c µ− and | − | | − | 0 setting τ = µ(1+σ)2/3/γ (note that τ < 1), we get that

Y (x) 1 5/2 k > (4.13) 1 k Yk 1(x) 1+ γc− τ − 0 Recalling (4.6) and (4.13) above, we obtain

1 Yk 1 1 c q (x) 1 c2 qk−1(x) r (x) = Y (x)e 2 2 k > Y (x)e 2 Yk−1 k 2 k 2 k Y Y 1 k k Y (x) 1 − Yk−1 1 1 Yk−1 k c2qk−1 > Yk 1(x) Yk 1(x)e 2 Yk 1(x) 2 − 2 − − Yk 5/2 1 Y 1 1 − Yk−1 k > Y (x) [r (x)] Yk−1 1 k k 1 k 1 1+ γc− τ 2 − − 0

37 Now, we have that

Y 1 5/2 1 k < 1 < cτ k 1 k − Yk 1 − 1+ γc− τ − 0 2/3(1+σ) for some positive constant c. Also, we assumed that x p > c µ− , so that | − | 0 2/3(1+σ) 5/2 5/3(1+σ)k x q > x p >c0µ− so that Yk 1(x)= yq(x) > ǫc0 µ− . Combining | − | | − | − these, we get

5/2 cτ k Y 1 ǫ 5/2 k 5/3(1+σ)k Yk−1 rk(x) > c0 µ− [rk 1(x)] 1 k − 1+ γc0− τ 2 h i and the conclusion follows by considering the separate cases rk 1(x) 1 and rk 1(x) < − ≥ − 1.

The immediate corollary is

Corollary 29. Either ∞ rk(x) > cl (4.14) 0 Y or ∞ rk(x) > cl r0(x) (4.15) 0 ! Y 5/2 cτ l 1 ǫ 5/2 5/3(1+σ)l for cl = −1 l 2 c0 µ− 1+γc0 τ h i h i Since 0∞ cl > 0, we get (c) of lemma 18 Q 4.5 Proof of Theorem 17(A), (B) and (C)

Note that (2.4) and our choice of Weierstrass data gives that

2 ∂zHk KΣk (z)= −| 4 | . (4.16) cosh Vk For p m , it is clear that F (p) = (0, 0,p) for all k. Thus, for k>l we can then ∈ l k estimate l µ− ∂zHk(p) > 4 (4.17) | | ak

38 2 since Vk(x, 0) = 0 for all x R and hence AΣ (p) . For p M M , there is ∈ | k | →∞ ∈ \ ∞ l a sequence of points p m such that p p <γ− j . We then get lj ∈ lj | − lj | l l µ− j µ− j ∂zHlj (p) > 2 2 > 2l 2 2 (4.18) | | ((p pl )+ a ) (γ− j + a ) − j lj lj k 2 Taking j and ak < γ− gives that AΣ (p) and proves (A) of Theorem → ∞ | lj | →∞ 17. Since V (x, y) > 0 for y = 0, we see that x (n(x, y)) = 0, and hence Σ is graphical k 6 3 6 k away from the x axis, which proves (C) of Theorem 17. 3 − Now, for δ > 0 set S = z dist(Re z, M) > δ . From (2.4), it is immediate that δ { | }

2 supk supΩ S AΣk (z) < (4.19) k\ δ | | ∞ for any δ > 0. This combined with Heinz’s curvature estimate for minimal graphs gives (B)

4.6 Proof of Theorem 17 (D) and The Structure Of The Limit Lamination

Lemma 30. A subsequence of the embeddings F : Ω R3 converges to a minimal k k → lamination of the cylinder C

d Proof. Let K be a compact subset of Ω . Then for p K, we have that supk Hk(z) < ∞ ∈ | dz | . Montel’s theorem then gives a subsequence converging smoothly to a holomor- ∞ phic function on K. By continuity of integration this gives that the embeddings F : K R3 converge smoothly to a limiting embedding. Thus the surfaces Σ k → k converge to a limit lamination of B that is smooth away from the x axis. r0 3 − Let I = (t , t ) R be an interval comprising the compliment of the M in R 1 2 ⊂ and consider ΩI = Ω Re z I . Then ΩI is topologically a disk, and by lemma ∞ ∩{ ∈ } 30, the surfaces Σ F (Ω ) are contained in t < x < t R3 and converge k,I ≡ k I { 1 3 2} ⊂ to an embedded minimal disk ΣI . Now, Theorem 17(C) (which we have already 1 2 established), gives that ΣI consists of two multi-valued graphs ΣI , ΣI away from

39 the x axis. We will show that each graph Σj is valued and spirals into the 3 − I ∞ − x = t and x = t planes as claimed. { 3 1} { 3 2} Note that by (2.8) and (C), the level sets x = t Σj for t

limk (cos Uk(t, 0), sine Uk(t, 0), 0) (4.20) →∞ −

t2 t1 First, suppose t m for some l. Then we get that, for any k>l and any t< − . 1 ∈ l 2 t1+2t t1+2t l ds − Uk(t1 +2t, 0) Uk(t1 + t, 0) = ∂sUk(s, 0)ds>c2µ 2 2 2 − t1+t t1+t ((s t1) + ak) Z Z − (4.21) (by the Cauchy-Reimann equations U = V ). Then, since a 0 as k , we k,x k,y k → →∞ get that

t1+2t l l ds c2µ− limk Uk(t1 +2t, 0) Uk(t1 + t, 0) >c2µ− > (4.22) →∞ − (s t )4 64t3 Zt1+t − 1 and hence t + t< x < t +2t contains an embedded N -valued graph, where { 1 | 3| 1 } t c µ l N > 2 − . (4.23) t 64t3 Note that N as t 0 from above and hence Σ spirals into the plane x = t . t →∞ → I { 3 1} Now, suppose that t1 / M . Then there exists a sequence of non-negative integers ∈ ∞ l j l l such that t p <γ− j , with p m . Then set t = t + γ− j and consider the j 1 − lj lj ∈ lj 1 intervals j+1 j Ij = [t , t ]. (4.24)

Note that for j large I I. Then, for k>l and s I we may estimate j ⊂ j ∈ j lj lj c2µ− c2µ− ∂sUk(s, 0) > > (4.25) ((s p )2 + a2)2 (4γ 2lj + a2)2 − lj k − k l since s p < 2γ− j . We then get − lj lj lj 1 lj j j+1 c2µ− c2µ− (1 γ− )γ− Uk(t , 0) Uk(t , 0) > Ij − (4.26) 2lj 2 2 2lj 2 2 − | |(4γ− + ak) ≥ (4γ− + ak) Taking limits, we get

1 3 lj j j+1 c2(1 γ− ) γ limk Uk(t , 0) Uk(t , 0) > − (4.27) →∞ − 16 µ 40 Thus we see that tj+1 < x < tj Σj contains an embedded N -valued graph, where { 3 }∩ I j 1 3 lj c (1 γ− ) γ N 2 − (4.28) j ≈ 32π µ This again shows that Σj spirals into the plane x = t since as j , tj t , I { 3 1} → ∞ → 1 and N . Now for t int(M), every singly graphical component of F contained j →∞ ∈ lj l l in the slab t γ− j ,t < x < γ− j (by (4.28) there are many) is graphical over { − 3 } x3 = 0 Br (0) where lemma 27 gives. rl , which implies each component { } ∩ lj j → ∞ comverges to the plane x = t . This proves Theorem 17 (D). { 3 }

4.7 Appendix

Here we provide the computations that were omitted from section 4.3 proof of lemma 22. It suﬃces to show that h = u + iv is holomorphic on ω. Recall that z dz h(z)= (4.29) (z2 + a2)2 Z0 It is clear that the points ia lie outside of ω. Moreover, as ω is the subgraph of ± z dz a function, it is obviously simply connected, so that 0 (z2+a2)2 gives a well-deﬁned holomorphic function on ω. R

proof of lemma 23. We compute the real and imaginary components of (z2 + a2)2:

z2 + a2 = x2 y2 + a2 +2ixy, − z2 + a2 2 = x2 y2 + a2 2 4x2y2 +4ixy x2 y2 + a2 . − − − Set a = Re z2 + a2 2 = x2 y2 + a2 2 4x2y2, − − n o2 b = Im z2 + a2 =4xy x2 y2 + a2 , − n o and

2 2 c2 = z2 + a2 2 = a2 + b2 = x2 y2 + a2 2 4x2y2 + 16x2y2 x2 y2 + a2 2 . − − − n o 41 Now on ω (that is, on the set where y y (x)), we get the bounds | | ≤ 0 a (1 ǫ2)2 x2 + a2 2 4ǫ2(x2 + a2)2 = (1 ǫ2)2 4ǫ2 (x2 + a2)2, ≥ − − − − a (x2 + a2)2, ≤ b 4ǫ(x2 + a2)11/4 4ǫ(x2 + a2)2 ≤ ≤ since by assumption x ,a< 1 . Using that c2 = a2 + b2, | | 2 (1 ǫ2)2 4ǫ2 2 (x2 + a2)4 c2 1+16ǫ2 x2 + a2 4 . − − ≤ ≤ Recalling that

∂ 1 b ∂ 1 a u(x, y)= Im = − , v(x, y)= Re = ∂y 2 2 2 c2 ∂y 2 2 2 c2 (z + a ) (z + a ) We get

∂ 4 x p y up(x, y) | − || | ∂y ≤ 2 2 2 2 2 2 3 (1 ǫ ) 4ǫ ((x p) + a ) { − − } − and

∂ (1 ǫ2)2 4ǫ2 1 vp(x, y) { − − } ∂y ≥ 1+16ǫ2 ((x p)2 + a2)2 − If we restrict ǫ < ǫ0 for ǫ0 suﬃciently small we get that 4

(1 ǫ2)2 4ǫ2 { − − } >c . 1+16ǫ2 2

for constants c1 and c2. which immediately gives the lemma.

proof of corollary 24. Recalling deﬁnitions (4.2) and (4.3), we get

42 k Yk(x) l ∂ U (x, Y (x)) U (x, 0) µ− u (x, y) (4.30) | k k − k | ≤ ∂y l l=0 Z0 Xk Y k(x) l ∂ µ− up(x, y) ≤ 0 ∂y l=0 p ml Z X X∈ k yp(x) l ∂ µ− up(x, y) ≤ 0 ∂y l=0 p ml Z X X∈ k 2 l l c ǫ µ− (γ + 1) ≤ 1 Xl=0 and

k Yk(x)/2 ∂ min[Yk(x)/2,Yk(x)]Vk(x, y) vl(x, y) (4.31) ≥ 0 ∂y Xl=0 Z k Yk(x)/2 l ∂ > µ− v (x, y) ∂y p l=0 p m 0 X X∈ l Z k ǫc2 l Yk(x) 2 2 3/4 = µ− ((x p) + a )− 2 yp(x) − l=0 p ml c X X∈ = 2 q (x) 2 k proof of lemma 25. k l Yk(x) 2 2 3/4 qk(x) = ǫ µ− ((x p) + a )− (4.32) yp(x) − l=0 p m X X∈ l k 1 − l Yk(x) 2 2 3/4 ǫ µ− ((x p) + a )− ≥ yp(x) − l=0 p m X X∈ l k 1 − Yk(x) l Yk 1(x) 2 2 3/4 = ǫ µ− − ((x p) + a )− Yk 1(x) yp(x) − l=0 p m − X X∈ l Yk(x) = qk 1(x) Yk 1(x) − −

43 Chapter 5

Self-Shrinkers with a symmetry

5.1 Non-compact, rotationally symmetric, embedded, self-shrinking, asymptotically conical ends with positive mean curvature

We consider n-dimensional hypersurfaces Σn Rn+1 with boundary ∂Σ = , ⊆ 6 ∅ satisfying the self-shrinker equation for mean curvature ﬂow →x, →ν H = h i, (5.1) 2 away from ∂Σ, and normalize to extinction time T = 1. In the paper (18) (or is n > 2 another paper?), Huisken showed that the only positive mean curvature H > 0 rotationally symmetric surface Σn, deﬁned by revolution of an entire graph over the x1-axis is the cylinder. However, we will prove the following theorem.

Theorem 31. In Rn+1 there exists a 1-parameter family of rotationally symmetric, embedded, self-shrinking, asymptotically conical ends Σn with H > 0. Each is smooth n 1 n 1 n with boundary ∂Σ = S − a scaled S − in the R -hyperplane, centred at the origin and with diameter d< 2(n 1). − C Rn+1 In fact for each symmetricp cone in x1 0 with tip at the origin, { ≥ } ⊆ there is such a self-shrinker, lying outside of C , which is asymptotic to C as x . 1 →∞ 44 Remarks 1.

(1) Theorem 31 shows in particular that not all solutions will eventually stop being imbedded (becoming immersed only), for increasing (positive) x-values. This fact is an important and necessary part of understanding the question of uniqueness of embedded genus 0 self-shrinkers. Such uniqueness is thus a quite global property.

(2) It is interesting to note that such self-shrinking ends could potentially have applications in gluing constructions for self-shrinkers.

5.2 Initial conditions “at inﬁnity” for a non-linear ODE with cubic gradient term

The theorem we will prove is more precisely stated as follows:

Theorem 32 (:= Theorem 1’). For each ﬁxed ray from the origin,

rσ(x)= σx, rσ : (0, ) R , ∞ → + there exists a unique smooth graphical solution uσ : (0, ) R , of the rotational ∞ → + self-shrinker ODE

x n 1 u(x) 2 u′′(x)= u′(x)+ − 1+(u′(x)) (5.2) 2 u(x) − 2 h i satisfying uσ rσ, and which is O( 1 )-asymptotic to rσ as x + . Moreover any ≥ x → ∞ solution u : (0, ) R+ to 5.2 belongs to this family. ∞ → We will need the following lemma, which observes that, generally speaking, solutions become non graphical.

Lemma 33. Consider a ﬁxed ray from the origin rσ(x) = σx, where σ > 0. If √2 a ( , ), and (a, x∞) is a maximally extended graphical solution to the initial ∈ σ ∞ a

45 value problem x 1 u 2 u′′ = u′ + 1+(u′) , 2 u − 2  h i  u(a)= σa, (5.3)   u′(a) σ,  ≥ 3a  then xa∞ < 2 . 

Proof of Lemma 33. If we deﬁne for u the quantity x 1 u Φ(x) := u′ + , 2 u − 2 then by Equation (5.2) we have Φ(a)= 1 . We claim that in fact Φ(a) 1 always. σa ≥ σa Namely assuming this holds up to x we have

d x 1 u x u′ x x 1 u 2 u′ u′ + = u′′ = u′ + 1+(u′) dx 2 u − 2 2 − u2 2 2 u − 2 − u2 x 1 2h u′ i 1+(u′) 0, ≥ 2 σa − u2 ≥ under the assumption that a √2 . Hence the condition Φ(a) 1 is preserved by ≥ σ ≥ σa the ODE (as we move to the right, i.e. increasing x), and implies in particular u′ σ ≥ always. 1 2 Using this, we can estimate: u′′ 1+(u′) . Integrating this inequality gives ≥ σa x a u′(x) tan 2 − + arctan σ , ≥ σa h i which ﬁnally leads to σa 3a x < π arctan σ . ∞ 2 2 − ≤ 2

Lemma 34 (A useful bootstrapping identity). Any solution u : (0, ) R to (5.13) ∞ → satisﬁes the identity

2 ′ ∞ 1 ∞ (1 + u′(s) ) 1/2 R s z(1+u (z)2)dz u(x)=(n 1)x e− t dt ds, (5.4) − t2 u(s) Zx Zt

46 Proof. Suppose ﬁrst that we are given a solution u : (0, a) over an interval → ∞ (0, a). We can regard the solution u as solving the inhomogeneous linear equation determined by freezing the coeﬃcients of (??) at u:

2 2 x 2 u 1+(ϕ′ + σ) u′′ 1+ ϕ′ u′ + 1+ ϕ′ =(n 1) (5.5) − 2 2 − ϕ where we have set u = φ above. We simply solve the resulting linear equation with variable coeﬃcients, for x (0, a), by making the observation that a pair of spanning ∈ solutions of the homogeneous linear equation are

′ a R a z(1+ϕ 2)dz e s u (x)= x, and u (x)= x ds, (5.6) 1 2 s2 Zx Then computing the Wronskian W (s) = eFa(s), and matching the initial conditions in (5.13) gives

′ a R a z(1+ϕ 2)dz u(a) e s u(x) = x +(u(a) u′(a)a)x ds (5.7) a − s2 Zx a a 2 ′ 1 (1 + u′(s) ) 1/2 R s z(1+u (z)2)dz + +(n 1)x e− t dt ds, (5.8) − t2 u(s) Zx Zt The remainder of the proof will be concerned with asymptotic behavior of the terms in(5.7) as a . That is, we will show, using elementary arguments, that →∞ u(a) σ a → for some limit σ, and

′ a R a z(1+ϕ 2)dz e s (u(a) u′(a)a)x ds 0. − s2 → Zx Observe that for any solution u : (0, ) R the quantity Φ(x) = xu′(x) u(x) is ∞ → − always negative. For suppose Φ(x ) 0 for some point x . Then 1 ≥ 1

Φ′(x1)= x1u′′(x1) (5.9)

But 1 1 u′′(x1) > Φ(x1)+ > 0 (5.10) 2 u(x1)

47 so that once Φ is non-negative, it is strictly increasing. Moreover, one must also

have u′(x1) > 0 , since the positivity of Φ implies the positivity of u′. Thus, we will eventually have u> √2, Φ > 0, u′ >c on some interval [d, ) at which point we can ∞ apply Lemma 33 to show that u has to become non graphical. This then implies that ratio u(a) is monotonically decreasing in a, and hence converges to some limit σ 0. a ≥ The positivity of Φ also implies that

a a 2 ′ 1 (1 + u′(s) ) 1/2 R s z(1+u (z)2)dz u(x) > (n 1)x e− t dt ds − t2 u(s) Zx Zt This then gives that there exists an sequence ak increasing to inﬁnity such that u(a ) √2: Otherwise, one would have that u(x) < √2 for large enough x. Then k ≥ appealing to the inequality (5.10) gives the contraction

a a 2 ′ 1 (1 + u′(s) ) 1/2 R s z(1+u (z)2)dz 2 u(x) > (n 1)x e− t dt ds > = √2(n 1). − t2 u(s) √2 − Zx Zt

Moreover, we can choose such a sequence a so that u′′(a ) 0: If u′′ were positive k k ≥ only when u< √2, a glance at equation (5.2) gives that u would eventually be below

the line y = √2. For such a sequence ak, one has,

a ′2 1 2 2 1 2 2 ak 1/2 R k z(1+u(z) )dz ak/2 (s a ak s a e s e 2 − k ) e 2 − k − ds < + dt (5.11) s2 s2 s2 Zx Zx Zak/2 1 (1/4 1)a2 1 2 2 1 < e 2 − k + x − a a − a k k k Thus, taking this sequence a in (5.7), we derive the limiting expression for u k → ∞ as

2 ′ ∞ 1 ∞ s(1 + u′(s) ) 1/2 R s(1+u (z)2)dz u(x)=(n 1)x e− t dsdt + σx (5.12) − t2 u(s) Zx Zt

′ 2 1+u 2(x) Hereafter, we will denote the quantities 1/2x(1 + u′ (x)) and u by p(x, u) and g(x, u), respectively. When there is no chance of confusion, we will simply use

48 p(x) and g(x), respectively. With this notation, we may write 5.2 as p(x) u′′ p(x)u′ + u = g(x) − x and the identity 34 becomes

∞ 1 ∞ 1 R s p u(x)=2(n 1)x p(s)e− t dsdt − t2 u(s) Zx Zt As an immediate consequence, we see that u(x) σx. This then gives ≥

2(n 1) ∞ 1 ∞ R s p 2(n 1) u(x) σx < − p(x)e− t dsdt = − | − | σ t2 x Zx Zt and by similar reasoning 2(n 1) u′(x) σ < − | − | σx2 In particular, we see that any solution u : (0, ) R+ is monotonic over some ∞ → interval (d, ) for d > 0. We next show that u is actually monotonic over all of ∞ (0, ). ∞ To finish the proof, we find it illustrative to construct each solution uσ as a limit of approximating solutions. More specifically, fixing a σ 0, we solve the initial value ≥ problem

x n 1 u 2 u′′ = u′ + − 1+(u′) , 2 u − 2  h i  u(a)= aσ, (5.13)   u′(a)= σ.  for a positive. On any interval where the solution exists, one derives the analogous  identity

a a 2 ′ 1 (1 + u′(s) ) 1/2 R s z(1+u (z)2)dz u(x)=(n 1)x e− t dt ds, (5.14) − t2 u(s) Zx Zt for x < a. The lack of terms in this expression corresponding to the homogeneous equation is a special property initial conditions. One derives analogous uniform estimates for solutions

2(n 1) u(x) σx < − | − | x

49 and 2(n 1) u′(x) σ < − | − | σx2 for any x < a. This then gives that each solution uσ,a extends to (0, a), and that the family u converges to a limiting solution u on (0, ). Note however, that { a}a>0 σ ∞ each approximate solution is really only approximate: lemma 33 implies that they do not remain graphical for values of x much larger than a. Proposition 1. For each σ, the map Φ given by

2 ′ ∞ 1 ∞ (1 + u′(s) ) 1/2 R s z 1+u (z)2 dz Φ(u)=(n 1)x e− t ( ) dt ds − t2 u(s) Zx Zt is a contraction

Proof. for two functions u1,u2 we may write ∞ ∞ s 1 1 1 R p2 Φ(u ) Φ(u ) = 2x p e− t (5.15) 2 − 1 t2 u − u 2 Zx Zt 2 1 ∞ ∞ s s 1 p2 R p2 R p1 + 2x e− t e− t t2 u − Zx Zt 1 ∞ ∞ s 1 1 R p1 + 2x (p p ) e− t t2 u 2 − 1 Zx Zt 1 = a + b + c

We estimate term a by u u a< 2|| 2 − 1|| σ2x2 Term c may be estimated by

1 ∞ 1 ∞ 1/4(t2 s2) 2 c< u′ u′ u′ + u′ se − dsdt = u′ u′ u′ + u′ || 2 − 1|||| 2 1||σ t2 || 2 − 1|||| 2 1||σx Zx Zt To estimate term b, we note that, for real numbers x,y

2 x ∞ 1 ∞ 2 2 1/4(t2 s2) b < u′ + u′ u′ u′ 1+ u′ 1/4(s t )e − dsdt || 2 1|||| 2 − 1|||| 2 ||σ t2 − Zx Zt 2 2x ∞ 1 ∞ 2 2 1/8(t2 s2) < u′ + u′ u′ u′ 1+ u′ 1/2(s t )e − sdsdt || 2 1|||| 2 − 1|||| 2 || σ t3 − Zx Zt 2 2x ∞ 1 ∞ τ = u′ + u′ u′ u′ 1+ u′ τe− dτdt || 2 1|||| 2 − 1|||| 2 || σ t3 Zx Z0 2 2 < u′ + u′ u′ u′ 1+ u′ || 2 1|||| 2 − 1|||| 2 ||σx

50 Here, all norms considered are sup norms over the interval (x, ) ∞

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54 Stephen James Kleene was born on July 21, 1982 in Rochester, New York. He attended Brighton High School in Brighton, New York. He graduated from the Uni- versity of Rochester with a Honors B.A. in Mathematics, earning Highest Distinction in Mathematics.