Geometry of Hypersurfaces

Appendix A

In this appendix we list standard deﬁnitions and well-known geometric facts for hypersurfaces in Rn+1. We consider only smooth embedded hypersurfaces M contained in some open set U ⊂ Rn+1. Let the embedding map be denoted by F : → Rn+1 with F() = M where ⊂ Rn is open. M is called a properly embedded hypersurface if F−1(K ) ⊂ is compact whenever K ⊂ U is compact. ∂ F The coordinate tangent vectors ∂i F(p) ≡ (p), 1 ≤ i ≤ n form a basis ∂pi of the tangent space Tx M at x = F(p) at every p ∈ . The metric on M is given by

gij = ∂i F · ∂ j F for 1 ≤ i, j ≤ n, the inverse metric by

ij −1 (g ) = (gij) and the area element of M by √ g = detgij. We are able to integrate compactly supported functions h : M → R over a properly embedded hypersurface. The integral is deﬁned by h ≡ hdHn ≡ h(x) dHn(x) = h(F(p)) g(p) dp. M M M Here Hn denotes the n-dimensional Hausdorff measure on M (see Appendix C). Note in particular that Hn(M ∩ K )<∞

109

K. Ecker, Regularity Theory for Mean Curvature Flow © Springer Science+Business Media New York 2004 110 Regularity Theory for Mean Curvature Flow for any compact K ⊂ U. The tangential gradient of a function h : M → R (which we may think of as a function on via the embedding map) is deﬁned by

M ij ∇ h = g ∂ j h ∂i F where we sum over repeated indices. For a smooth tangent vector ﬁeld

i ij X = X ∂i F = g X j ∂i F on M (note that Xi = X · ∂i F) we deﬁne the covariant derivative tensor by ∇ M j = ∂ j + j k = jl ∂ − k i X i X ik X g i Xl il Xk k where the Christoffel symbols ij are given by (∂ ∂ )T = k ∂ i j F ij k F (the superscript T denotes the tangential component of a vector) or in terms of the metric by 1 k = gkl ∂ g + ∂ g − ∂ g . ij 2 i jl j il l ij The tangential divergence of X on M is deﬁned by 1 √ div X =∇M X i = gij∇ M X = √ ∂ ggijX M i i j g i j and the Laplace–Beltrami operator of h on M by 1 √ h = div ∇ M h = gij ∂ ∂ h − k ∂ h = √ ∂ ggij∂ h . M M i j ij k g i j

For a smooth vector ﬁeld X : M → Rn+1 which is not necessarily tangent to M we can also deﬁne the divergence with respect to M by

ij divM X = g ∂i X · ∂ j F.

This reduces to the previous expression for tangent vector fields. Let ν be a choice of unit normal field to M. In particular, this satisfies

ν · ∂i F = 0 Appendix A. Geometry of Hypersurfaces 111

on M for 1 ≤ i ≤ n. Note also that ∂i ν is a tangent vector ﬁeld to M for 1 ≤ i ≤ n since ν has unit length. The second fundamental form of M is deﬁned by

Aij = ∂i ν · ∂ j F =−ν · ∂i ∂ j F.

The eigenvalues κ1,...,κn of the Weingarten map (which is a map from the tangent space to itself) given by i = ik A j g Akj are called the principal curvatures of M. The mean curvature can then be expressed in various forms by

n = κ = i = ij = ij∂ ν · ∂ = ν. H i Ai g Aij g i j F divM i=1 The mean curvature vector of M is given by

H =−Hν.

One readily checks the identity M F = H. (A.1)

In the special case where M = graph u, u : → R (that is F(p) = (p, u(p)), the upward unit normal vector ν is given by (−Du, 1) ν = (A.2) 1 +|Du|2 and the mean curvature of M by Du −H = div (A.3) 1 +|Du|2 where D and div denote the gradient and the divergence on Rn. For graphs with radial symmetry, that is, when u(x) = f (r) with r =|x|, the identity (A.3) reduces to − − = 1 f + n 1 H f (A.4) 1 + ( f )2 1 + ( f )2 r 112 Regularity Theory for Mean Curvature Flow where the denotes derivatives of f with respect to r. The Riemann curvature tensor of M is deﬁned by

∇ M ∇ M −∇M ∇ M = M l i j Xk j i Xk Rijkl X (A.5) where X is a tangent vector ﬁeld on M and

M M M M M ∇ ∇ ≡∇ ∇ −∇ M i j τi τ j ∇τ τ i j denotes the Hessian operator in a local orthonormal frame τ1,...,τn. The Riemann tensor satisﬁes the symmetry relations

M =− M Rijkl R jikl and

M =− M . Rijkl Rklij The Gauss equations express this tensor in terms of the second fundamental form of M by

M = − . Rijkl Aik A jl Ail A jk The Codazzi equations state that the 3-tensor of covariant derivatives of the second fundamental form

∇ M = (∇ M ) A i A jk is totally symmetric. One defines covariant derivatives and Hessian and Laplacian operator of tensor fields analogously as in the case of vector fields. In an orthonormal τ ,...,τ ∇ M ∇ M frame 1 n, we denote for instance the component of i j A with τ τ ∇ M ∇ M respect to k and l by i j Akl. We now wish to calculate the Laplacian of the second fundamental form and its squared norm

| |2 = i j = ij kl . A A j Ai g g Aik A jl To simplify notation, we work in a local orthonormal frame so that we are allowed to use lower indices only. We also use subscripts for derivatives. The distinction between tensor components and covariant derivatives will always Appendix A. Geometry of Hypersurfaces 113

∇ M be clear from the context. For instance, Aijk stands for k Aij and Aijkl for ∇ M ∇ M l k Aij. Formula (A.5) reads as − ≡∇M ∇ M −∇M ∇ M = M = M Xijk Xikj k j Xi j k Xi Rkjil Xl Xl Rlijk in this notation after some rewriting using the symmetries of the Riemann tensor. The analogous identity for the second fundamental form (or any other 2-tensor) appears in the convenient and easily remembered form − = M + M . Aijkl Aijlk Aim Rmjkl Amj Rimkl (A.6) We now calculate (recall that we are still summing over repeated indices)

M Aij = Aijkk = Aikjk = Akijk = + M + M Akikj Akm Rmijk Ami Rmkjk

= Akkij + Akm(Amj Aik − Amk Aij) + Ami(Amj Akk − Amk Akj) 2 = Hij −|A| Aij + HAik Akj

2 where |A| = Aij Aij and H = Akk. Note that we have used the Codazzi equations to obtain the second and the ﬁfth identity and (A.6) for the fourth identity. In the ﬁfth step, we have also used the Gauss equations (for the details of this computation we essentially followed [S1] ). In standard notation, this becomes =∇M ∇ M + k −| |2 M Aij i j H HAik A j A Aij (A.7) which is usually referred to as the Simons’ identity (see [Si]). Contracting (A.7) with

ij ik jl A = g g Akl yields the Bochner type formula | |2 = ij∇ M ∇ M + |∇ M |2 + i k − | |4 M A 2 A i j H 2 A 2 HAij Ak A j 2 A (A.8) |∇ M |2 ∇ M = (∇ M ) where A denotes the squared norm of the tensor A k Aij . We shall also need an expression for the Laplacian of the unit normal vector ﬁeld. In order to derive this, it is convenient to work with geodesic normal coordinates on M, that is, assume

gij = δij 114 Regularity Theory for Mean Curvature Flow and

T (∂i ∂ j F) = 0

(the superscript T stands for the tangential component) at the point x = F(p) ∈ M where we do the calculation. In these coordinates we can simply write

M ν = ∂i ∂i ν

(of course summed over repeated indices) at the point x = F(p). Moreover,

Aij =−∂i ∂ j F · ν,

∂i ν = Aij∂ j F and the Codazzi equations read as

∂i Aij = ∂ j Aii.

We then compute at this point ∂i ∂i ν = ∂i Aij∂ j F = ∂i Aij∂ j F + Aij∂i ∂ j F = ∂ j Aii∂ j F − Aij Aijν using the Codazzi equations in the last step. Thus we arrive at

2 M M ν = −|A| ν +∇ H (A.9)

(see also [S1] or [EH1]). This identity is often referred to as the Jacobi field equation. Derivatives of functions and vector fields on hypersurfaces can also be defined in terms of projections from Rn+1 onto the tangent space of M. This is the framework used in geometric measure theory where coordinate systems are not available. The notions defined below can be adapted to countably n- rectifiable subsets of Rn+1. ∈ Rn+1 → For x M we define the projection pTx M : Tx M by (w) = w − (ν( ) · w)ν( ). pTx M x x

Let f : U → R be differentiable where U is an open subset of Rn+1 containing M. We could alternatively consider f : M → R and require that it has Appendix A. Geometry of Hypersurfaces 115 a differentiable extension into an open neighbourhood of M in Rn+1 (recall that derivatives of f in the direction of tangent vectors to M are independent of the particular extension of f ). We deﬁne the tangential gradient of f with respect to M by ∇ M ( ) = ( ( )) = ( ) − ν( ) · ( )ν( ) f x pTx M Df x Df x x Df x x for x ∈ M where Df(x) denotes the usual gradient of f in Rn+1 (or of its extension into Rn+1). This can also be written as

n ∇ M ( ) = ( )τ f x Dτi f x i i=1 ( ) τ where Dτi f x denotes the directional derivative of f with respect to i and τ1,...,τn form an orthonormal basis of Tx M. For a differentiable vector ﬁeld X : U → Rn+1 (or X : M → Rn+1) the derivative in the direction w ∈ Rn+1 is given by ⎛ ⎞ w1 ∂ Xi ⎜ . ⎟ Dw X(x) = (x) ⎝ . ⎠ . ∂ . x j 1≤i, j≤n+1 wn+1 The tangential divergence of X with respect to M is then deﬁned as

divM X(x) = divRn+1 X(x) − ν(x) · Dν(x) X(x). (A.10)

Alternatively,

n ( ) = τ · ( ) = ( · ( )) divM X x i Dτi X x trace pTx M DX x i=1 which is equivalent to the intrinsic formulation given earlier. The dot is used both for the dot product of vectors and composition of linear maps. It is usually clear from the context which one is meant. The Laplace–Beltrami operator on M of a twice differentiable function f is deﬁned by

M M f = divM ∇ f.

From the identity H =−Hν =−(divM ν)ν 116 Regularity Theory for Mean Curvature Flow we calculate using (A.10)

M f = divM Df − divM ((ν · Df)ν) = divM Df − Hν · Df (A.11) 2 = divM Df + H · Df = Rn+1 f − D f (ν, ν) + H · Df where

2 D f (ν, ν) ≡ ν · Dν Df is the second derivative of f in normal direction to M. If f (x) = xi for 1 ≤ i ≤ n + 1, then one checks using (A.11) that M xi = H · ei

th n+1 (ei is the i basis vector in R ) which implies the identity M x = H (see (A.1) above). The divergence theorem for smooth, properly embedded hypersurfaces with smooth boundary states that for any C1-vector ﬁeld X : M¯ → Rn+1 the identity divM X =− H · X + X · γ (A.12) M M ∂ M holds where γ denotes the outer unit normal ﬁeld to ∂ M, which is tangent to M at all boundary points. Of course, we are integrating over ∂ M with respect to the standard surface element or equivalently (n − 1)-dimensional Hausdorff measure. If X has compact support or if ∂ M =∅this reduces to divM X =− H · X. (A.13) M M φ ∈ 2(Rn+1) For a function C0 the divergence theorem implies divM Dφ =− H · Dφ (A.14) M M or equivalently

M φ = 0 (A.15) M Appendix A. Geometry of Hypersurfaces 117

φ ∈ 2(Rn+1) η ∈ 2(Rn+1) in view of (A.11). For C0 and C one also checks that M M φM η =− ∇ φ ·∇ η = ηM φ. (A.16) M M M If φ does not vanish on the boundary of M, then M divM Dφ =− H · Dφ + ∇ φ · γ (A.17) M M ∂ M where we have used that

Dφ · γ =∇M φ · γ since γ is tangent to M.

We will also make use of the following version of the coarea formula: Let u : M → R be a proper Lipschitz continuous function on M, that is, assume that the sets {x ∈ M, u(x)

Note that by Sard’s theorem, M ∩ ∂ Br is a smooth hypersurface of M for almost every r > 0. We also assume that Br ∩ ∂ M =∅. Appendix B

Derivation of the Evolution Equations

In this appendix we give derivations of the evolution equations for most of the geometric quantities used in the earlier chapters. These were ﬁrst derived in [Hu1], see also [GH] for the case of curves. We will only have to calculate the time derivative of these quantities and then combine this with the identities for the Laplacian stated in Appendix A. For ease of notation during our calculations, we will denote time derivatives by ∂t . We start with expressions involving the metric and calculate

∂t gij = 2 ∂t ∂i F · ∂ j F = 2 ∂i ∂t F · ∂ j F (B.1)

where we have used the fact that coordinate derivatives of F(p, t) commute. In view of the mean curvature ﬂow equation

∂t F =−Hν

(see 2.2) this becomes

∂t gij = 2 ∂i (−Hν) · ∂ j F =−2 H∂i ν · ∂ j F

since ν · ∂ j F = 0 for 1 ≤ j ≤ n. This implies

∂t gij =−2 HAij. (B.2)

119 K. Ecker, Regularity Theory for Mean Curvature Flow © Birkhäuser Boston 2004 120 Regularity Theory for Mean Curvature Flow

One easily checks that the inverse metric and the area element then satisfy

ij ij ∂t g = 2 HA (B.3) and √ √ √ 2 2 ∂t g =−H g = −|H| g (B.4) since the derivative of g is given by

ij ∂t g = gg ∂t gij.

Note that (B.1) also implies the more general formula √ √ √ ∂ F ∂ g = ggij∂ ∂ F · ∂ F = g div . (B.5) t i t j Mt ∂t We continue with the time derivative of the second fundamental form. For convenience, we will again calculate in geodesic normal coordinates on Mt as in Appendix A. That is we assume

gij = δij and

T (∂i ∂ j F) = 0

(T stands for the tangential component) at the point x = F(p, t) ∈ Mt where we carry out the calculation. Since

Aij =−∂i ∂ j F · ν we compute at x = F(p, t),

∂t Aij =−∂t (∂i ∂ j F · ν)

=−∂i ∂ j ∂t F · ν − ∂i ∂ j F · ∂t ν

=−∂i ∂ j ∂t F · ν using that ∂t ν is tangent and ∂i ∂ j F has no tangential component. We substi- tute ∂t F =−Hν and again use ∂ j ν · ν = 0 to obtain

∂t Aij = ∂i ∂ j H + H∂i ∂ j ν · ν Appendix B. Derivation of the Evolution Equations 121 and therefore

∂t Aij = ∂i ∂ j H − H∂i ν · ∂ j ν.

Since in normal coordinates

∇ Mt ∇ Mt = ∂ ∂ i j H i j H at x = F(p, t) we conclude that

∂ =∇Mt ∇ Mt − k . t Aij i j H HAik A j (B.6)

ij The identities (B.3), (B.6) and H = g Aij, therefore yield ∂ − = | |2. t Mt H H A (B.7) One also easily derives the evolution equation for Aij and then checks the identity

∂ | |2 = ij∇ Mt ∇ Mt + k ij t A 2 A i j H 2 HAik A j A (B.8) which combined with (A.8) implies the evolution equation ∂ − | |2 = | |4 − |∇ Mt |2. t Mt A 2 A 2 A (B.9) For a derivation of the evolution equation for the mth covariant derivative of A we refer to [Hu1]. It remains to calculate the evolution of ν. Since ∂t ν is a tangent vector we can express it in terms of the coordinate tangent vectors ∂ j F by

ij ∂t ν = g ∂t ν · ∂ j F ∂i F.

From this we infer

ij ij ∂t ν =−g ν · ∂ j (−Hν)∂i F = g ∂ j H∂i F where we have used (2.2), the product rule and the identity ν · ∂ j F = 0. By deﬁnition of the tangential gradient this implies

Mt ∂t ν =∇ H. (B.10)

In combination with (A.9), identity (B.10) yields the evolution equation ∂ − ν =| |2ν. t Mt A (B.11) 122 Regularity Theory for Mean Curvature Flow

It is then an easy exercise to check that the gradient function

v = 1 ν · en+1 satisﬁes

|∇ Mt v|2 ∂ − v = −|A|2v − 2 (B.12) t Mt v see [EH1].

For the convenience of the reader we append the

Proof of the Weak Maximum Principle (Proposition 3.1). One first notes that if h(p, t) satisfies ∂ − ≤ ·∇Mt , t Mt h a h then for any >0 the function h˜ = h − t which agrees with h at time 0 satisfies ∂ − − ·∇Mt ˜ < . t Mt a h 0 (B.13) ˜ At a point where for the first time maxMt h reaches a value larger than maxM0 h, the standard derivative criteria at local maxima say that ˜ ˜ ∂t h ≥ 0,∂i h = 0 and ˜ (∂i ∂ j h) ≤ 0

≤ , ≤ ∇ Mt for 1 i j n. By the deﬁnition of and Mt given in Appendix A, ∂ − − ·∇Mt ˜ ≥ t Mt a h 0 at this point. This contradicts (B.13). Hence h˜ is bounded by the initial values of h at all positive times. Letting → 0 we see that the same applies to h. Appendix C

Background on Geometric Measure Theory

In this appendix we list some of the basic definitions and facts from geometric measure theory. We will concentrate on notions which we actually use in the main part of the book. For more detailed information especially of a mo- tivational nature we refer to a number of excellent texts such as [S1], [EG1], [Mor] and [R]. The standard reference is of course [F]. We start by recalling the definition of m-dimensional Hausdorff measure on RN . Let A ⊂ RN , 0 ≤ m < ∞ and δ>0 and define ∞ m ∞ m diam C j Hδ (A) = inf ωm , A ⊂ C j , diam C j ≤ δ j=1 2 j=1

N where diam denotes the diameter of a subset of R and ωm equals the volume of the m-dimensional unit ball when m is an integer and any convenient positive constant otherwise. We then deﬁne the m-dimensional Hausdorff measure of A by

m m m H (A) = lim Hδ (A) = sup Hδ (A). δ 0 δ>0 Hausdorff measures have a number of nice properties. They are Borel regular (see [S1]) which allows us to approximate the measure of an arbitrary Hm- measurable set A from above by the measure of open sets containing A and from below by closed sets contained in A. Moreover, Hausdorff measures are invariant under isometries of Euclidean space and exhibit the natural scaling behaviour. On Rm, the measure Hm

123 K. Ecker, Regularity Theory for Mean Curvature Flow © Birkhäuser Boston 2004 124 Regularity Theory for Mean Curvature Flow agrees with the standard m-dimensional Lebesgue measureand for m-dimensional submanifolds of RN it corresponds to the standard m-dimensional area. We will also consider Radon measures on RN , that is, Borel regular measures μ which satisfy the additional condition that μ(K )<∞ for every compact K ⊂ RN . Note that Hm is not a Radon measure except when m = N. We will however often consider the restriction of the Hausdorff measure Hn to properly embedded hypersurfaces M of Rn+1, that is, μ deﬁned by

μ(A) = Hn(M ∩ A) for A ⊂ Rn+1, which is a Radon measure. For a locally Hn-integrable function f on M, another Radon measure is deﬁned by μ(A) = fdHn. M∩A

N A sequence (μ j ) of Radon measures R is said to converge to a limiting measure μ if

lim φ dμ j = φ dμ j→∞

φ ∈ 0(RN ) for all C0 . As a consequence of the Borel regularity of the measures one concludes that

lim μ j (A) = μ(A) j→∞ holds for any bounded Borel set A as long as

μ(∂ A) = 0

(see [Gi], Appendix A). By the ﬁniteness of μ on compact subsets this implies that for every x ∈ RN ,

lim μ j (Bρ(x)) = μ(Bρ(x)) (C.1) j→∞ for all but countably many ρ>0. We will also use the m-dimensional density of a set A ⊂ RN at a point x ∈ RN with respect to Hausdorff measure. This is deﬁned by

m H (A ∩ Bρ(x)) m(A, x) = lim , ρ m 0 ωmρ Appendix C. Background on Geometric Measure Theory 125 whenever this limit exists. One checks, for instance, that if M is a smooth, properly embedded hypersurface of Rn+1, then

n(M, x) = 1 at all points x ∈ M. We will also make use of the following

Lemma C.1 If A ⊂ RN is Hm-measurable and satisﬁes Hm(A)<∞, then m(A, x) = 0 for Hm–almost every x ∈ RN \A.

For a proof we refer to [S1]. This uses in particular another fact we need in Chapter 5.

Proposition C.2 (Vitali Covering Theorem) Let F be a collection of non- degenerate closed balls B¯ in RN and assume that there is a ﬁxed constant which bounds all their radii. Then one can select a countable disjoint family ¯ F of balls Bρ j in such that ∞ ¯ ⊂ ¯ . B B5ρ j B¯ ∈F j=1

We furthermore need the following fact which appears in the proof of the monotonicity formula in [I2].

Lemma C.3 (μ-Integrability of Gaussians) Let μ be a Radon measure on N N R . Let x0 ∈ R and assume that there are positive constants A0, R0 and p so that

p μ(BR(x0)) ≤ A0 R for all R ≥ R0. Then for every γ>0 2 −γ |x−x0| e dμ(x) ≤ c(p, A0, R0,γ)<∞ RN and 2 −γ |x−x0| e dμ(x) ≤ δ(p, A0,γ, R) N R \BR (x0) where δ(p, A0,γ, R) → 0 as R →∞. 126 Regularity Theory for Mean Curvature Flow

Proof. We may set x0 = 0. Pick a radius R > max{1, R0} and abbreviate B = B (0). We then calculate R R 2 e−γ |x| dμ(x) Rn+1 ∞ 2 2 = e−γ |x| dμ(x) + e−γ |x| dμ(x) + + Rn 1∩B = Rn 1∩ (B + \B ) R j 1 R j 1 R j ∞ p p( j+1) −γ R2 j ≤ A0 R + R e < ∞. j=1 This implies the result.

Finally, we introduce generalisations of smooth hypersurfaces, so-called countably n-rectifiable subsets of Rn+1. These are defined as subsets M ⊂ Rn+1 which satisfy ∞ M ⊂ N j j=0 n 1 n+1 where H (N0) = 0 and N j is an embedded C -hypersurface in R for every j ≥ 1. We will mainly work with the following property of countably n-rectifiable subsets (for a proof see [S1], chapter 3). Proposition C.4 (Approximate Tangent Spaces) Any countably n-rectifiable subset M ⊂ Rn+1 which is also Hn-measurable and satisfies Hn(M ∩ K )<∞ for all compact K ⊂ Rn+1 admits an approximate tangent space at Hn– almost every x ∈ M. This means that the rescales M x,λ = λ−1(M − x), λ > 0 n+1 of M about x converge to a hyperplane Tx M ⊂ R in the sense of Radon measures, that is, lim φ dHn = φ dHn λ0 x,λ M Tx M φ ∈ 0(Rn+1) for all C0 . At points where Tx M exists one checks that n(M, x) = 1. Appendix D

Local Results for Minimal Hypersurfaces

In this appendix we consider smooth, properly embedded minimal hypersurfaces, that is, hypersurfaces M ⊂ Rn+1 with zero mean curvature. Since we present only local results inside balls intersecting M, we shall tacitly assume that these balls do not meet the boundary of M. The main topics are two versions of regularity theorems in the spirit of Al- lard’s and de Giorgi’s theorem ([All], [DG], [S1]), stated for smooth minimal hypersurfaces. These results are analogues of the local regularity theorems for mean curvature ﬂow discussed in Chapter 5. Their proofs are based on standard monotonicity and mean value formulas for minimal hypersurfaces whose derivation we include. For more comprehensive information and for related material we refer to the books by Simon ([S1]) and by Colding and Minicozzi ([CM]).

Theorem D.1 (Mean Value Formula) Consider a smooth function f on a ( ) ⊂ Rn+1 smooth, properly embedded minimal hypersurface M in Bρ0 x0 . Then the formula |( − )⊥|2 d 1 = d x x0 f f + dr r n ∩ ( ) dr ∩ ( ) |x − x |n 2 M Br x0 MBr x0 0 1 + r 2 −|x − x |2 f n+1 0 M 2r M∩Br (x0)

holds in the distributional sense for r ∈ (0,ρ0) (and also classically for almost every r ∈ (0,ρ0)). If furthermore f ≥ 0 and f is subharmonic on M,

127 K. Ecker, Regularity Theory for Mean Curvature Flow © Birkhäuser Boston 2004 128 Regularity Theory for Mean Curvature Flow

that is, satisﬁes M f ≥ 0, then d 1 f ≥ . n 0 dr r M∩Br (x0)

If in addition x0 ∈ M we obtain as a consequence 1 f (x ) ≤ f (D.1) 0 ω ρn n M∩Bρ (x0)

n for all ρ ∈ (0,ρ0). Here ωn denotes the volume of the unit ball in R .

For f ≡ 1, Theorem D.1 reduces to the important monotonicity formula.

Corollary D.2 (Monotonicity Formula) Under the conditions of Theorem D.1, the formula d Hn(M ∩ B (x )) d |(x − x )⊥|2 r 0 = 0 (D.2) ω n | − |n+2 dr nr dr M∩Br (x0) x x0 holds in the distributional sense for any r ∈ (0,ρ0) (and classically for almost every r ∈ (0,ρ0)). In integrated form it states that Hn( ∩ ( )) Hn( ∩ ( )) |( − )⊥|2 M Bρ x0 − M Bσ x0 = x x0 ω ρn ω σ n | − |n+2 n n M∩(Bρ (x0)\Bσ (x0)) x x0 (D.3) for all 0 <σ ≤ ρ ≤ ρ0. If x0 ∈ M, then Hn( ∩ ( )) M Bρ x0 ≥ n 1 (D.4) ωnρ for every ρ ∈ (0,ρ0).

Proof. For simplicity we set x0 = 0 and Br = Br (0). By the coarea formula (A.19) d 1 f = f (D.5) |∇ M | | dr M∩Br M∩∂ Br x for almost every r ∈ (0, R) (and also in the distributional sense). By Sard’s theorem, we may assume that M ∩∂ Br is a smooth hypersurface inside M for Appendix D. Local Results for Minimal Hypersurfaces 129

these r. The divergence theorem (A.12) applied to M∩Br and our assumption that H ≡ 0 on M imply x x divM f = f · γ (D.6) M∩Br r M∩∂ Br r where ∇ M | | T γ = x = x . |∇ M |x|| |x T |

Here x T denotes the component of x tangential to M and we have used that

x T ∇ M |x|= . |x| One easily checks that x · γ = |∇ M |x|| r and therefore, by combining (D.5) and (D.6), d 1 − |∇ M |x||2 x f = f + div f . (D.7) |∇ M | || M dr M∩Br M∩∂ Br x M∩Br r Using again the coarea formula and the identities

|x⊥|2 1 − |∇ M |x||2 = |∇⊥|x||2 = |x|2 and x n x div f = f + ·∇M f, M r r r this yields d d |x⊥|2 n x T f = f + f + ·∇M f. | |2 dr M∩Br dr M∩Br x r M∩Br M∩Br r Since 1 x T =− ∇ M (r 2 −|x|2) 2 130 Regularity Theory for Mean Curvature Flow

2 2 and r −|x| vanishes on ∂ Br , we may integrate by parts in the last integral to obtain d d |x⊥|2 n 1 f = f + f + (r 2 −|x|2) f. | |2 M dr M∩Br dr M∩Br x r M∩Br 2r M∩Br Finally, we multiply this identity by r −n and combine terms to arrive at d 1 d |x⊥|2 1 f = f + (r 2 −|x|2) f. n | |n+2 n+1 M dr r M∩Br dr M∩Br x 2r M∩Br Note that here we have also used the coarea formula again to infer 1 d |x⊥|2 d |x⊥|2 f = f . n | |2 | |n+2 r dr M∩Br x dr M∩Br x

If now f ≥ 0 and f is subharmonic on M, that is, satisﬁes M f ≥ 0, then d 1 f ≥ . n 0 dr r M∩Br (x0) In particular 1 lim f r→0 ω n nr M∩Br (x0) exists. If in addition x0 ∈ M, then the continuity of f and smoothness of M imply that this limit equals f (x0). This yields the mean value inequality 1 f (x ) ≤ f 0 ω ρn n M∩Bρ (x0) for all ρ ∈ (0,ρ0).

As an example, we consider the function given by ( ) = ( − )2 . f x x x0 n+1

By (A.1) in Appendix A we have M xn+1 = 0. One checks using the product rule that ( − )2 = |∇ M ( − )2 |2 ≥ . M x x0 n+1 2 x x0 n+1 0

We may therefore apply (D.1) above with ρ replaced by ρ/2, x0 replaced ( ) = ( − )2 ∈ ∩ ( ) by x and f x x x0 n+1 at every point x M Bρ/2 x0 . Note that Bρ/2(x) ⊂ Bρ(x0) for these points. This yields Appendix D. Local Results for Minimal Hypersurfaces 131

Corollary D.3 (Distance to Hyperplane) Let M be a smooth, properly em- ( ) ⊂ Rn+1 ρ ∈ ( ,ρ ) bedded minimal hypersurface in Bρ0 x0 . Then for any 0 0 the estimate c(n) (x − x )2 ≤ (x − x )2 sup 0 n+1 n 0 n+1 ρ ρ M∩B (x0) M∩Bρ (x0) 2 holds.

Remark D.4 (Tilt-excess) The L2-integral in the corollary also controls an expression termed tilt-excess which plays a crucial role in Allard’s and de Gi- orgi’s regularity theorem for minimal hypersurfaces. For simplicity, let x0 = 0. We deﬁne the tilt excess of M inside Bρ = Bρ(0) by 1 M 2 1 2 Exc (M, 0,ρ) ≡ |∇ x + | = 1 − ν . ρn n 1 ρn n+1 M∩Bρ M∩Bρ If we multiply the equation 2 = |∇ M 2 |2 M xn+1 2 xn+1 η2 η ∈ 2( ) η ≡ , ≤ η ≤ |∇ M η|≤ by where C0 Bρ satisﬁes 1 on Bρ/2 0 1 and 2/ρ, integrate by parts, use Young’s inequality and rearrange terms, we obtain |∇ M |2η2 ≤ 2 |∇ M η|2. xn+1 4 xn+1 M M The properties of η then imply the Cacciopoli type inequality ( ) 1 M 2 c n 2 Exc (M, 0,ρ/2) = |∇ x + | ≤ x . (ρ/ )n n 1 ρn+2 n+1 2 M∩B ρ M∩Bρ 2 Alternatively, we could have used the divergence theorem (A.13)

divM X = 0 M 2 n+1 with X = xn+1en+1η . Here en+1 is the standard basis vector in R . Note that

M M 2 divM (xn+1en+1) =∇ xn+1 · en+1 = |∇ xn+1| in view of the identity Dxn+1 = en+1. The remainder of the argument follows as before. 132 Regularity Theory for Mean Curvature Flow

Recall that minimal hypersurfaces are stationary solutions of mean curvature flow. Thus the following local estimate can be regarded as a special case of the higher derivative bounds in Proposition 3.22 for mean curvature flow. Since the proof is not significantly easier in the minimal surface case (one essentially has to replace the heat operator by the negative of the Laplacian everywhere), we will not include the proof here.

Proposition D.5 (Smoothness Estimate) Let M be a smooth, properly embedded minimal hypersurface in Rn+1 which satisfies c sup |A(x)|2 ≤ 0 ρ2 M∩Bρ (x0) for some radius ρ and constant c0. Then for every m ≥ 1 there is a constant cm depending on n, m and c0 such that c sup |∇m A(x)|2 ≤ m . ρ2(m+1) M∩B ρ (x0) 2 The next theorem is the minimal surface analogue of White’s local regularity Theorem 5.6 for mean curvature flow [W3]. It easily extends to limits of smooth, properly embedded minimal hypersurfaces and can be regarded as a simplified version of Allard’s and de Giorgi’s regularity theorem (see [All], [DG] and [S1]). We basically follow the proof in [W3] but use the above local smoothness estimates rather than standard regularity theory for elliptic partial differential equations.

Theorem D.6 (Regularity Theorem A) Let M be a smooth, properly em- n+1 bedded minimal hypersurface in R . There are constants 0 > 0 and c0 > 0 depending only on n as well as a constant α = α(n, 0) ∈ (0, 1) such that, whenever Hn( ∩ ( )) M Bρ x0 ≤ + n 1 0 ωnρ holds at some point x0 ∈ M and for some radius ρ>0, then c sup |A|2 ≤ 0 . ρ2 M∩Bαρ (x0)

For the proof of the theorem we need the following technical lemma which is an easy consequence of the monotonicity formula (see [S1]). Appendix D. Local Results for Minimal Hypersurfaces 133

Lemma D.7 Let M be a smooth, properly embedded minimal hypersurface in Rn+1. Suppose that for some >0 the inequality Hn( ∩ ( )) M Bρ x0 ≤ + n 1 ωnρ

n+1 holds at some point x0 ∈ R and for some radius ρ>0. Then there exists a constant β = β(n, ) such that Hn( ∩ ( )) M Bσ y ≤ + n 1 2 ωnσ for every y ∈ Bβρ (x0) and every σ ∈ (0,(1 − β)ρ).

Proof. Using the monotonicity formula at y ∈ Bβρ (x0) as well as the inclu- sion B(1−β)ρ(y) ⊂ Bρ(x0) we estimate Hn( ∩ ( )) Hn( ∩ ( )) Hn( ∩ ( )) M Bσ y ≤ M B(1−β)ρ y ≤ M Bρ x0 n n n ωnσ ωn((1 − β)ρ) ωn((1 − β)ρ) 1 ≤ (1 + ) ≤ 1 + 2 (1 − β)n if β is chosen sufﬁciently small depending on n and .

Proof of Theorem D.6. For simplicity set x0 = 0 and Br = Br (0). In view of Lemma D.7, it sufﬁces to prove the following. There are constants 0 and c0 depending only on n such that whenever a smooth, properly embedded minimal hypersurface M ∈ Rn+1 satisﬁes 0 ∈ M and Hn( ∩ ( )) M Bσ y ≤ + n 1 0 ωnσ for all y ∈ Bβρ and σ ∈ (0,(1 − β)ρ), then

2 2 sup σ sup |A| ≤ c0. σ ∈(0,βρ) M∩Bβρ−σ

This implies the desired curvature estimate with α = β/2 by choosing σ = βρ/2. (Note from the proof of the lemma that β(n, ) → 0 for → 0.) Scaling so that βρ = 1 we have to verify the following statement. There exist constants 0 > 0 and c0 > 0 depending only on n as well as β = 134 Regularity Theory for Mean Curvature Flow

β(n, 0) ∈ (0, 1) such that whenever a smooth, properly embedded minimal hypersurface M ∈ Rn+1 satisﬁes 0 ∈ M and Hn( ∩ ( )) M Bσ y ≤ + n 1 0 ωnσ

−1 for all y ∈ B1 and σ ∈ (0,β − 1), then the estimate

2 2 sup σ sup |A| ≤ c0 σ ∈(0,1) M∩B1−σ holds. Suppose this is not true. Then there exists a sequence (M j ) of smooth, properly embedded minimal hypersurfaces with 0 ∈ M j and a sequence (β j ) → 0 such that Hn( ∩ ( )) M j Bσ y ≤ + 1 n 1 (D.8) ωnσ j ∈ σ ∈ ( ,β−1 − ) for all y B1 and 0 j 1 , but γ 2 ≡ σ 2 | |2 →∞. j sup sup A (D.9) σ ∈(0,1) M j ∩B1−σ

We can then ﬁnd radii σ j ∈ (0, 1) for which γ 2 = σ 2 | |2 j j sup A ∩ −σ M j B1 j ¯ and points y j ∈ M j ∩ B1−σ so that γ 2 = σ 2| ( )|2. j j A y j

From this and the deﬁnition of γ j one checks that

2 2 sup |A| ≤ 4|A(y j )| . ∩ − σ / M j B1 j 2 ( ) ⊂ Since Bσ j /2 y j B1−σ j /2 this implies

2 2 sup |A| ≤ 4|A(y j )| . (D.10) ∩ σ / M j B j 2 Appendix D. Local Results for Minimal Hypersurfaces 135

˜ We now rescale our sequence of minimal hypersurfaces by setting M j ≡ λ−1( − ) λ =| ( )|−1. ˜ j M y j with j A y j Then M j satisﬁes ˜ 0 ∈ M j , |A(0)|=1 as well as the estimate

sup |A|2 ≤ 4. ˜ ∩ M j Bλ−1σ / j j 2

λ−1σ = γ →∞ Since j j j we conclude

sup |A|2 ≤ 4 (D.11) ˜ M j ∩BR for every R > 0 and for all sufﬁciently large j depending on R. Rescaling (D.8) about y = y j with σ = λ j R yields

Hn( ˜ ∩ ) ≤ M j BR ≤ + 1 1 n 1 ωn R j σ <β−1 − for all sufﬁciently large j depending on R to ensure that j 1. For the lower bound on the area ratio we have used (D.4) at y ∈ M and then rescaled. In view of the local smoothness estimates for minimal hypersurfaces in ˜ Proposition D.5 we may select a subsequence of (M j ) (again denoted by ˜ n+1 (M j )) which converges smoothly on compact subsets of R to a complete, smooth, properly embedded minimal hypersurface M ⊂ Rn+1. ˜ Passing the above properties of M j to limits, we see that 0 ∈ M with |A(0)|=1 and Hn( ∩ ) M BR = n 1 ωn R holds for all R > 0. The monotonicity formula (D.3) therefore implies that x⊥ = 0 for all x ∈ M , which says that M is a minimal cone with vertex at 0 ∈ M . Since M is smooth it therefore has to be a hyperplane. However, this contradicts the fact that |A(0)|=1.

Finally, we present a second local regularity theorem which is the analogue of Theorem 5.7. 136 Regularity Theory for Mean Curvature Flow

Theorem D.8 (Regularity Theorem B) Let M be a smooth, properly em- n+1 bedded minimal hypersurface in R . There exist constants 0 > 0 and c0 > 0 depending only on n such that if 1 sup (x − x )2 ≤ σ n+2 0 n+1 0 σ ∈(0,ρ) M∩Bσ (x0) holds at some point x0 ∈ M and for some radius ρ>0, then c sup |A|2 ≤ 0 . ρ2 M∩B ρ (x0) 8

Again, we ﬁrst require a technical result which is an easy consequence of Lemma D.9.

Lemma D.9 Let M be a smooth minimal hypersurface in Rn+1 satisfying ( − )2 ≤ σn+2 x x0 n+1 M∩Bσ (x0) for some x0 ∈ M and some σ>0. Then there exists a constant c(n) such that

( − )2 ≤ ( ) σ 2 sup x y n+1 c n M∩B σ (y) 4 for all y ∈ M ∩ Bσ/4(x0).

Proof. For y ∈ Bσ/4(x0) we have Bσ/4(y) ⊂ Bσ/2(x0). We may therefore apply Corollary D.3 with ρ replaced by σ at all points x ∈ M ∩ Bσ/4(y).

Proof of Theorem D.8. In view of Lemma D.9, it sufﬁces to prove the curvature estimate under the assumption that x0 ∈ M and

( − )2 ≤ σ 2 sup x y n+1 0 M∩B σ (y) 4 holds for all σ ∈ (0,ρ] and y ∈ M ∩ Bρ/4(x0). Setting again x0 = 0 and Br = Br (0) and scaling so that ρ = 4 we have to prove the following statement. There exist constants 0 > 0 and Appendix D. Local Results for Minimal Hypersurfaces 137

c0 > 0 depending only on n such that whenever a smooth, properly embedded minimal hypersurface M ⊂ Rn+1 satisﬁes 0 ∈ M and

( − )2 ≤ σ 2 sup x y n+1 0 M∩Bσ (y) for all σ ∈ (0, 1] and y ∈ M ∩ B1, then the estimate

2 2 sup σ sup |A| ≤ c0 σ ∈(0,1) M∩B1−σ (0) holds. We proceed exactly as in the proof of Theorem D.6. In fact, an identi- cal scaling argument yields a family of smooth, properly embedded minimal ˜ ˜ hypersurfaces (M j ) which satisfy 0 ∈ M j , |A(0)|=1, the curvature estimate

sup |A|2 ≤ 4 ˜ M j ∩BR as well as the height bound R sup |yn+1|≤ ˜ j M j ∩BR for all R > 0 and sufﬁciently large j depending on R. The complete, smooth, properly embedded minimal hypersurface M ⊂ n+1 ˜ R arising as limit of a subsequence of (M j ) therefore satisﬁes yn+1 = 0 for all y ∈ M . This implies the contradicting properties M = Rn+1 ×{0} and |A(0)|=1at0∈ M . Appendix E

Remarks on Brakke’s Clearing Out Lemma

In this appendix we prove a version of Brakke’s clearing out lemma which resembles more closely his original one ([B], Lemma 6.3). It is stronger than the version presented in Proposition 4.23 in that it links the size of the cleared out set to the clearing out time. Our proof here uses an isoperimetric inequality on hypersurfaces which is a consequence of the Michael–Simon Sobolev inequality. This resembles in some sense Brakke’s original argument. We restrict ourselves to the case where n = 2 as in this case the exposi- tion is particularly clean. This is because the local square integral of mean curvature controls the evolution of local area under mean curvature ﬂow; see Corollary 4.7. The general case can be found in [Ec1]. The argument there was suggested by John Hutchinson. As in Appendix D, we shall only consider local results for hypersurfaces M inside balls. We therefore tacitly assume that these balls do not meet the boundary of M.

We begin with a statement of the Michael–Simon Sobolev inequality (see [MS] and [S1]) and some of its consequences.

Proposition E.1 (Michael–Simon Inequality) Let M be a smooth, properly embedded hypersurface in Rn+1. Let f be a nonnegative, compactly sup-

139 K. Ecker, Regularity Theory for Mean Curvature Flow © Birkhäuser Boston 2004 140 Regularity Theory for Mean Curvature Flow ported Lipschitz continuous function on M. Then the inequality

n−1 n n n M n f n−1 d H ≤ c(n) |∇ f |+|H| f d H M M holds.

Approximating the characteristic function of a ball Br (x0) by Lipschitz functions (noting that by Sard’s theorem M ∩ ∂ Br (x0) is a smooth hypersurface inside M for almost every r > 0) we obtain as a consequence the following version of the isoperimetric inequality. A more general version of this inequality appears in [All].

Corollary E.2 (Isoperimetric Inequality) Let M be a smooth, properly n+1 embedded hypersurface in R . Then there exists a constant c0 depending n+1 only on n such that for any x0 ∈ R and almost every r > 0, − n n 1 n n H (M ∩ Br (x0)) n ≤ c0 H (M ∩ ∂ Br (x0)) + |H| dH . M∩Br (x0) An important consequence of the isoperimetric inequality is the following lower bound on the area ratio of a hypersurface inside a ball whenever the integral of |H|n taken over this ball is sufﬁciently small.

Lemma E.3 (Lower Area Ratio Bound) Let M be a smooth, properly em- n+1 n+1 bedded hypersurface in R . Suppose for some ball Bρ(x0) ⊂ R and ∈ ( , −1) some 0 c0 (here c0 is the isoperimetric constant) the conditions 1/n |H|nd Hn < M∩Bρ (x0) and

n H (M ∩ Bρ/2(x0)) > 0 hold. Then

n n H (M ∩ Bρ(x0)) ≥ θρ (E.1) n 1 where θ = (1 − c0 ) . 2nc0 Appendix E. Remarks on Brakke’s Clearing Out Lemma 141

Remark E.4 The positivity condition on the area of M inside Bρ/2(x0) ensures that the hypersurface enters Bρ(x0) sufﬁciently far. This is necessary as the pictures in Figure E.1 show. The left surface has small curvature integral and its area ratio is close to 1. The middle surface has small area ratio and large curvature integral. The curvature integral of the surface on the right is small, but it also has small area ratio since it hardly enters Bρ(x0) let alone intersects Bρ/2(x0).

Figure E.1: Surfaces inside a ball

Proof of Lemma E.3. For simplicity, set x0 = 0 and Br = Br (0). By the isoperimetric inequality in Corollary E.2, − n n 1 n n H (M ∩ Br ) n ≤ c0 H (M ∩ ∂ Br ) + |H| dH (E.2) M∩Br holds for almost every r > 0. Using the coarea formula (A.19) (we may assume that this holds for the same radii as (E.2)) and Holder’s¨ inequality we obtain 1/n − n−1 d 1Hn( ∩ ) n − | |n Hn ≤ Hn( ∩ ) c0 M Br 1 c0 H d M Br M∩Br dr in the distributional sense (and for almost every r > 0). Suppose now that 1/n |H|nd Hn < M∩Bρ and

n H (M ∩ Bρ/2)>0 142 Regularity Theory for Mean Curvature Flow

ρ> ∈ ( , −1) ∈ (ρ/ ,ρ) hold for some 0 and 0 c0 . Then for r 2 we infer d Hn(M ∩ B )1/n ≥ γ dr r in the distributional sense, where 1 γ = (1 − c0 ) . nc0 Integrating this inequality over the interval (ρ/2,ρ) yields the desired lower bound.

We are now in the position to prove Brakke’s clearing out lemma. We restrict ourselves to the case n = 2. For the general case we refer the reader to [Ec1].

Proposition E.5 (Clearing Out Lemma) Consider a smooth, properly em- n+1 bedded solution (Mt )t>0 of mean curvature ﬂow in R . There exists an absolute constant c > 0 such that if M0 satisﬁes

2 2 H (M0 ∩ Bρ(x0)) ≤ ρ

3 for some x0 ∈ R ,ρ>0 and >0, then

H2(M ∩ B ρ (x )) = 0 t 4 0 for t = c ρ2.

Remark E.6 This proposition says that if a ball initially contains a piece of surface with relatively small area ratio, then the ball of one quarter the radius will eventually become empty when this surface is evolved by mean curvature. The time it takes to clear out the ball is proportional to the square of its radius and to the initial area ratio. However, it is possible that later the evolving surface re-enters this ball as the sequence in Figure E.2 shows.

Proof of Proposition E.5. By scaling and translating we may assume that x0 = 0 and ρ = 1. We then proceed in the following way. Let

2 3 ϕ(x, t) = (1 −|x| − 4t)+ Appendix E. Remarks on Brakke’s Clearing Out Lemma 143

Figure E.2: Surface re-entering small disk a long time after clearing out be the shrinking test function considered in Remark 4.8. This satisﬁes the inequality d ϕ ≤− |H|2ϕ (E.3) dt Mt Mt in view of Corollary 4.7 and (4.4). In particular, 2 ϕ ≤ ϕ ≤ H (M0 ∩ B1) ≤ Mt M0 which yields

2 H (Mt ∩ B1/2) ≤ 8 (E.4) for all t ≤ 1/16, in view of the fact that ϕ ≥ 1/8 in B1/2 for those t. We want to show that for small enough δ>0 the inequality d ϕ ≤−δ (E.5) dt Mt holds for all t ≤ 1/16. For ≤ δ/16 this would then imply

2 H (Mt ∩ B1/2) = 0 when t = /δ, hence the result. Suppose (E.5) does not hold. We may then also assume that

2 H (Mt ∩ B1/4)>0 144 Regularity Theory for Mean Curvature Flow for all t ≤ 1/16 since otherwise we would again be ﬁnished. Then for any δ>0 there exists a time t ≤ 1/16 for which |H|2 ≤ 8 |H|2ϕ ≤ 8δ. Mt ∩B1/2 Mt Using (E.1) in Lemma E.3 with ρ = 1/2 we obtain

2 H (Mt ∩ B1/2) ≥ θ/4 where √ 2 θ = 1 − δ . 2 1 c0 8 16c0 This obviously contradicts (E.4) for ≤ δ/16 and small enough δ. Appendix F

Local Monotonicity in Closed Form

In this appendix, we present an outline of the proof of the local monotonicity formula in Remark 4.18 (3). More details can be found in [Ec3] where we also establish a corresponding mean value formula. Let us first repeat some of the definitions given in Remark 4.18 (3). n+1 For r > 0 we define a heat-ball Er = Er (0, 0) centred at (0, 0) ∈ R × R to be the bounded open set

n+1 n+1 Er ={(x, t) ∈ R × R, t < 0,ψr (x, t)>0}⊂R × R

where

n ψr ≡ log ( r ) ≡ ψ + n log r.

Since |x|2 n 4πt ψ (x, t) = − log(− ) r 4t 2 r 2 we can also write = ×{ } Er BRr (t) t −r2/4π

145 K. Ecker, Regularity Theory for Mean Curvature Flow © Birkhäuser Boston 2004 146 Regularity Theory for Mean Curvature Flow

2 Note that Rr (−r /4π) = Rr (0) = 0 and 2 n max Rr (t) = Rr (−r /4πe) = r −r2/4π≤t≤0 2πe so that in particular

√ 2 Er ⊂ B n × (−r /4π,0). 2πe r We introduce the integral quantity 2 2 A (M ∩ Er ) ≡ |∇ψ| +|H| ψr M∩Er where we adopt the shorthand notation ∇=∇Mt and 0 f ≡ fdμt dt. M∩ − r2 ∩ Er 4π Mt BRr (t) In [Ec3], a certain amount of work goes into showing that A (M ∩ ) ≤ ( )Hn( ∩ √ ). Er c n M− r2 B 2n 4π π r

This estimate ensures finiteness of A (M ∩ Er ) for all r ∈ (0, R) as long as the space-time track M of a solution of mean curvature flow is well-defined inside

√ 2 B 2n × (−R /4π,0) π R in that it does not have a boundary inside this set and the condition Hn( ∩ √ )<∞ M−R2/4π B 2n π R is satisﬁed.

We first require three integration formulas for M. Note that the first two do not use mean curvature flow but hold for the space-time track of a general family of smooth, properly embedded hypersurfaces as long as all integrals involved are finite. They are direct consequences of the divergence theorem and the coarea formula. The third one uses the evolution equation for the area element from Lemma 4.1. For simplicity of notation we use ∇ and for ∇ Mt and Mt respectively. Appendix F. Local Monotonicity in Closed Form 147

Proposition F.1 (Integration formulas for heat-balls) Let M denote the space-time track of a family (Mt ) of smooth, properly embedded hypersurfaces and let f be an arbitrary function defined on M. Then the identities d n f ψr = f, (i) dr M∩Er r M∩Er d n |∇ψ|2 =− ψ (ii) dr M∩Er r M∩Er hold in the distributional sense for r ∈ (0, R) (and classically for almost every r ∈ (0, R)) as long as all integral expressions are finite. If the family (Mt ) is a solution of mean curvature flow, then the additional identity ψ d 2 = |H| ψr (iii) M∩Er dt M∩Er holds for all r ∈ (0, R).

We will illustrate the simple integration by parts and coarea formula in- gredients of the proof by a not quite rigorous argument. For this purpose we assume that all quantities are differentiable and that ∩∂ ∈( , ) ∈(− 2/ π, ) Mt BRr (t) is a smooth submanifold of Mt for r 0 R and t r 4 0 . This is not usually the case as some simple examples in [Ec3] show. The detailed proof of this proposition given in [Ec3], however only uses that Mt ∩ ∂ ∈ BRr (t) has zero n-dimensional measure for all except countably many r (0, R) and for almost all t ∈ (−r 2/4π,0) for these r. This in turn follows from the ﬁniteness assumption on the above integral expressions. Let us ﬁrst establish identity (iii): The coarea formula (A.19) with u(x) = |x| and r replaced by Rr (t) and Lemma 4.1 yield ∂ d 1 n−1 ψr dμt = ψr d H Rr (t) dt ∩ ∩∂ |∇|x|| ∂t Mt BRr (t) Mt BRr (t) d 2 + ψr − |H| ψr . ∩ dt ∩ Mt BRr (t) Mt BRr (t) ψ ∂ The boundary integral disappears since r vanishes on BRr (t). Integrat- ing over (−r 2/4π,0) implies (iii) since

2 Rr (−r /4π) = Rr (0) = 0. 148 Regularity Theory for Mean Curvature Flow

Similarly ∂ d 1 n−1 f ψr = f ψr d H Rr (t) dr ∩ ∩∂ |∇|x|| ∂r Mt BRr (t) Mt BRr (t) ∂ψ + f r . ∩ ∂r Mt BRr (t) Again the boundary integral is zero and since ∂ψ d n r = (n log r) = ∂r dr r 2 identity (i) follows after integrating over (−r /4π,0) and noting that Rr (t) vanishes at the endpoints of this interval. The divergence theorem yields n−1 ψ dμt = ∇ψ · γ d H ∩ ∩∂ Mt BRr (t) Mt BRr (t) γ ∩ ∂ ψ(·, ) =− where is the outward unit normal to Mt BRr (t). Since t n log r ∂ ψ(·, )>− on BRr (t) and t n log r in BRr (t) we ﬁnd that ∇ψ γ =− . |∇ψ| Hence we obtain ∇ψ · γ d Hn−1 =− |∇ψ| d Hn−1. ∩∂ ∩∂ Mt BRr (t) Mt BRr (t) ={ ∈ Rn+1, −ψ( , )< } Since BRr (t) x x t n log r , the coarea formula (A.18) with u =−ψ(·, t) and r replaced by n log r yields d d |∇ψ|2 = |∇ψ| d Hn−1 (n log r). dr ∩ ∩∂ dr Mt BRr (t) Mt BRr (t)

This implies (ii), again after integrating with respect to t and using that Rr (t) vanishes at the endpoints of the time interval (−r 2/4π,0).

Next, we shall require the following evolution equations for ψ = log . Note that the expression in identity (iii) below agrees with the differential Harnack expression in Euclidean space ([Ha3], [LY]). 150 Regularity Theory for Mean Curvature Flow

Proof. The main work was done in the proof of the integration formulas in Proposition F.1. The evolution equation for ψ, Lemma F.2 (i) yields d |∇ψ|2 = − + ψ −| −∇⊥ψ|2 +||2 . Mt H H M∩Er M∩Er dt We now use identities (i)–(iii) in Proposition F.1, with f =|H|2 in identity (i), and rearrange terms to obtain 2 2 r d 2 2 |∇ψ| +|H| ψr = |∇ψ| +|H| ψr M∩ n dr M∩ Er Er + |H −∇⊥ψ|2 M∩Er Multiplying both sides by nr−(n+1) and combining terms implies the theorem.

Finally we want to derive some of the properties of A (M∩ Er ) discussed in Remark 4.18.

Remark F.4 (1) Since |x|2 n 4πt R (t)2 −|x|2 ψ (x, t) = − log − = r r 4t 2 r 2 −4t the integral quantity can be written as 0 | T |2 ( )2 −| |2 x 2 Rr t x A (M ∩ Er ) = +|H| (x, t) dμt dt. − r2 ∩ 4t2 −4t 4π Mt BRr (t) n If Mt = R for every t < 0 this reduces to the expression 0 |x|2 dx dt − r2 ( ) 4t2 4π BRr t considered in [Fu], [EG2], [Wa] which satisﬁes for every r > 0 1 0 |x|2 0 |x|2 dx dt = dx dt = 1. r n − r2 ( ) 4t2 − 1 ( ) 4t2 4π BRr t 4π BR1 t

(2) For M ≡{(Mt , t), t < 0} we deﬁne the rescaled ﬂow Mr ≡{( r , ), < } Ms s s 0 Appendix F. Local Monotonicity in Closed Form 151

r = −1 where Ms r Mr2s. One readily checks the scaling behaviour A (M ∩ E ) r = A(Mr ∩ E ). r n 1 In particular, if (0, 0) is a smooth point of M, then Mr → Rn × R− for r → 0 and therefore A (M ∩ ) Er n − lim = A((R × R ) ∩ E1) = 1 r→0 r n by (1) above. (3) In view of Proposition F.1 (i) and Lemma F.2 (ii) we have the alternative expressions ψ 2 d A (M ∩ Er ) = |∇ψ| + M∩ dt Er n = + H ·∇⊥ψ − |∇⊥ψ|2 − M∩Er 2t which simplify to n 0 n A (M ∩ ) = = Hn( ∩ ) Er Mt BRr (t) dt M∩ −2t − r2 −2t Er 4π

on homothetically shrinking solutions. This expression for A (M ∩ Er ) is used to prove the following fact.

Proposition F.5 Let M be a homothetically shrinking solution of mean curvature ﬂow. Then for all r > 0 and all t < 0 we have A (M ∩ E ) r = dμ . n t r Mt

Proof. A homothetically shrinking solution satisﬁes Mr = M for all r > 0 −n so that in particular r A (M ∩ Er ) is independent of r in view of Remark = −1 > < F.4 (2). Since this says that Mt r Mr2t for all r 0 and t 0, we infer by setting r 2 =−1/t that √ Mt = −tM−1 (F.1) 152 Regularity Theory for Mean Curvature Flow for all t < 0. Huisken’s monotonicity formula (Theorem 4.11) states that 1 − |x|2 μ = μ = 4 μ d t d −1 n e d −1 ( π)2 Mt M−1 4 M−1 for all t < 0 on homothetic solutions. It therefore sufﬁces to prove the identity | |2 − x √ 4 μ = A (M ∩ ). e d −1 E 4 π M−1 By Remark F.4 (2), 0 n A (M ∩ √ ) = Hn ∩ √ . E 4 π Mt B 2nt log(−t) dt (F.2) −1 −2t We now use Fubini’s theorem to calculate ∞ | |2 | |2 − x n − x e 4 d μ−1 = H {x ∈ M−1, e 4 ≥ τ} dτ M−1 0 1 n √ = H − ∩ τ. M 1 B −4 log τ d 0

n When τ = (−t) 2 , | |2 0 − x n n n √ e 4 d μ− = (−t) 2 H M− ∩ B dt. 1 − 1 −2n log(−t) M−1 −1 2t By scaling we obtain √ n n √ n √ (− ) 2 H − ∩ = H − − ∩ t M 1 B −2n log(−t) tM 1 B 2nt log(−t) which establishes the result in view of (F.1) and (F.2). Appendix F. Local Monotonicity in Closed Form 149

Lemma F.2 The function ψ satisﬁes the evolution equation d + ψ = −|∇ψ|2 −|H −∇⊥ψ|2 +|H|2. (i) dt Mt

This takes the equivalent form

dψ n + |∇ψ|2 + = H ·∇⊥ψ − |∇⊥ψ|2. (ii) dt 2t If moreover M is a homothetically shrinking solution of mean curvature ﬂow, ⊥ that is, if it satisﬁes H −∇ ψ = 0 on each Mt , then we have

dψ n + |∇ψ|2 + = 0. (iii) dt 2t

Proof. All equations can be derived from (3.6) and Lemma 4.1 in combination with the identities x Dψ(x, t) = 2t and n ψ = + H ·∇⊥ψ. 2t Since ψ = log , equation (i) also follows with the help of the chainrule from the evolution equation (4.6) for .

We are now ready to prove the monotonicity formula for A (M ∩ Er ). This can in some sense be regarded as an integrated version of the evolution equation for ψ in Lemma F.2 (i).

Theorem F.3 Under the ﬁniteness conditions stated at the beginning of the Appendix, the formula d A (M ∩ E ) n r = |H −∇⊥ψ|2 n n+1 dr r r M∩Er holds in the distributional sense for r ∈ (0, R) (and also in the classical sense for almost every r ∈ (0, R)). Bibliography

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Allard’s and de Giorgi’s regularity Brakke solution, 4, 49, 56, 72, 84 theorem, 4, 86, 127, 131, Brakke’s clearing out lemma, 29, 132 74, 89, 102, 139, 142 Angenent torus, 13, 29 Brakke’s regularity theorem approximate tangent space, 82, local version, 4, 87, 90, 91, 102, 126 102 area continuity hypothesis, 83, 85, main version, 4, 81, 83, 84, 99 87 area decay, 48 version in a ball, 83 local, 51 area element, 9, 47, 109 Cacciopoli type inequality, 131 evolution of, 48 Cantor process, 86 area functional, 9 catenoid, 86 area growth, 54 change of variables, 62, 94, 103 area ratio, 74, 89 Christoffel symbols, 110 lower bound, 74, 140 clearing out lemma, 29, 74, 89, uniform bound, 98 102, 139, 142 upper bound, 75 coarea formula, 117, 128, 141, Arzela–Ascoli theorem, 46, 73, 95 146 Codazzi equations, 112 backward heat kernel, 52 compact solution, 18, 19, 48 blow-up, 13, 71, 90 comparison principle, 18 homothetically contracting, cone, 13, 16, 17, 21 72, 85 cone comparison, 28 multiple density, 87 continuous rescaling, 73 smoothness of, 72 convex set, 27 Bochner type formula, 113 coordinate system, 114 Borel regular, 123 coordinate tangent vectors, 109 boundary, 116, 139 countably n-rectiﬁable set, 81, 82, Brakke ﬂow, 4 114, 126

159 160 INDEX covariant derivative, 110 global gradient, 32 curvature, 30 height, 18 evolution equation, 38 interior, 3, 90 local estimate, 39, 90, 91, local, 3, 23 132, 136 local curvature, 39, 95, 132, mean, 111 136 principal, 111 local gradient, 31 Riemann tensor, 112 Schauder, 90 curve shortening flow, 20 smoothness, 39, 90, 95, 132, cylinder comparison, 29 135 evolution equation, 119 de Giorgi–Nash–Moser theory, 40 for higher derivatives, 38 density, 89, 102, 124, 126 for area element, 120 n-dimensional, 82, 102, 124, for backward heat kernel, 56 126 for curvature, 38 Gaussian, 69, 87, 89, 90, 92, for distance, 26, 28 102 for gradient function, 31, 122 multiple, 87 for graph, 15 properties, 125 for inverse metric, 120 unit, 87 for logarithm of heat kernel, differential Harnack inequality, 148 58, 148 for mean curvature, 121 directional derivative, 115 for metric, 119 distance estimate, 27, 77 for unit normal, 121 divergence, 110 extrinsic distance, 20 tangential, 110 theorem, 50, 115, 116, 129, first singular time, 2, 81, 85, 88 131, 146 first variation formula, 48 dumbbell, 23, 29 weighted version, 49 Fubini’s theorem, 152 embedding, 7, 109, 110 proper, 109 entire graph solution, 3, 32 Gauss equations, 112 estimate Gaussian, 59, 61, 105 distance, 27 μ-integrability of, 125 distance to a hyperplane, 131 Gaussian density, 61, 65, 69, 87, for dimension of singular set, 89, 90, 92, 102 2 examples, 61 INDEX 161

locally defined, 65 subsolutions of, 76 lower bound, 70 heat-ball, 66, 145 of cylinder, 61 height estimate, 18 of hyperplane, 61 Hessian operator, 112 of smooth solution, 63 of tensor fields, 112 of sphere, 62 homothetic solution, 3, 4, 10, 61 upper-semicontinuity, 70 contracting, 12, 13, 53, 68, generalised solution, 3, 72 72, 74, 151 geometric measure theory, 81, expanding, 12, 16, 17, 21 114, 123 hyperboloid comparison, 28 good point, 89, 99, 102 hyperplane, 77, 82 gradient flow, 9 L2-deviation from, 4, 77, 90, gradient function, 30, 122 102 evolution equation, 31 distance from, 77, 131 global estimate, 32 double density, 87, 90 local estimate, 31 Gaussian density of, 61 graph solution, 12, 13, 16–18, 34, graph over, 46 68 hypersurface, 4 entire, 3, 19, 79 compact, 19 evolution equation, 15 convex, 19 grim reaper, 16, 31 embedded, 109 generalised, 49, 126 geometry of, 109 Hamilton’s Ricci flow graph, 14 programme, 1 initial, 19, 21, 28, 29, 54, 85 harmonic map heat flow, 89 limiting, 74, 99 Hausdorff measure, 47, 109, 123 mean curvature flow of, 7 heat equation, 8, 21 minimal, 8, 28, 86, 89, 127 backward, 58 properly embedded, 30, 109, linear, 21 124 non linear, 8 Holder’s¨ inequality, 141 representation formula, 59 heat kernel, 57 backward, 52, 53 initial condition, 17 logarithm of, 67, 145, 148 initial data, 16, 20 heat operator, 26 initial hypersurface, 19, 21, 28, comparison, 26 29, 54, 85 on Euclidean space, 26 integral varifold, 56, 86, 87 on mean curvature flow, 26 integration by parts, 56, 117, 130 162 INDEX interior estimate, 90 vector, 7, 15, 56, 111 intrinsic distance, 20 zero, 86, 87, 127 isoperimetric inequality, 139, 140 mean curvature flow, 7 Brakke solution, 49 limit points, 27 Jacobi field equation, 114 normalised, 58, 73 stationary solutions, 132 Kato’s inequality, 41, 44 weak formulation, 49 with additional forces, 90 mean value formula Laplace–Beltrami operator, 8, 24, for minimal hypersurfaces, 25, 110, 115 127 of second fundamental form, mean value inequality 112 for mean curvature flow, 76 of tensor fields, 112 for minimal hypersurfaces, of unit normal, 113 127, 130 Lebesgue measure, 124 measure level-set flow, 4, 84 n Hδ -, 100 limiting solution, 2 Borel regular, 123 linear growth, 16, 17 convergence, 124 local area bound, 52 Hausdorff, 109, 123 local estimate, 23 Lebesgue, 124 localisation Radon, 106, 124 function, 34 metric, 109 spherically shrinking, 64 inverse, 109 via cut-off functions, 63 Michael–Simon inequality, 139 locally finite measure, 81, 82 minimal cone, 13, 135 minimal graph, 40 maximum principle, 10, 19, 23 minimal hypersurface, 1, 8, 28, localised version, 35 86, 89, 127 noncompact version, 24, 32, minimum principle, 24 76, 79 monotonicity formula, 53 strong version, 85 and Gaussian density, 61 weak version, 23, 122 and weighted first variation, mean curvature, 8, 14 58 bounded, generalised, 85, 86 for general kernels, 57 nonnegative, 13 for minimal hypersurfaces, positive, 85 66, 128 INDEX 163

Huisken’s, 53 Perelman’s work on Ricci flow, 1, local version in closed form, 5 66, 145, 150 points reached by solution, 27 localised version, 55, 65 proper function, 117 normalised, 60 properly embedded, 30 rescaled local version, 65 properly embedded solution, 48 rescaled version, 62, 74 weighted, 55 Rademacher’s theorem, 117 Moser iteration, 76 Radon measure, 124 motion convergence, 124 by mean curvature, 12 rectifiability assumption, 81, 101 homothetic, 12 rectifiable set, 81, 82, 114, 126 normal, 12, 14 regular point, 81 radial, 12 regularity, 84 vertical, 14 almost everywhere, 84, 87 multiple density, 87 regularity theorem multiple density surface, 85 Allard’s and de Giorgi’s, 4, multiplicity, 86 86, 127 multiplicity function, 85 Brakke’s main version, 4, 81, 83, 87 for levelset flow, 84 neck, 23 local version, 4, 87, 90, 91 noncompact solution, 19, 27 local version for minimal nonsmooth solution, 81 hypersurfaces, 132, 136 normal coordinates, 113, 120 space-time version, 84 normal projection, 8 White’s local version, 4, 90 normal space, 8 regularity theory, 55, 81 rescaled solution, 2, 4, 13, 40, 62, orthonormal frame, 112 68, 71, 72, 88, 94, 103, 151 rescaling, 62, 65, 89 parabolic blow-up, 71 continuous, 73 partial differential equation, 15 limit, 71 elliptic, 16 parabolic, 62 parabolic, 15, 18, 23, 40, 85 sequence, 87, 88 standard methods for, 46 Ricci flow, 1, 40, 60 standard regularity theory, Riemann curvature tensor, 112 132 symmetry relations of, 112 164 INDEX

Sard’s theorem, 117, 128, 140 homothetically expanding, Schauder estimate, 90 12, 16, 17, 21 Schauder theory, 40 in the integral sense, 3, 49 second fundamental form, 111, integral varifold, 56 112 limiting, 2 self-similar solution, 2 noncompact, 19, 27 separation of variables, 1, 11 nonsmooth, 81 shrinking cylinder, 9, 12, 13, 61, of mean curvature flow, 7 85, 87 points reached by, 27, 81 shrinking sphere, 9, 10, 12, 31, properly embedded, 30, 48 61, 68 regularity of, 84 shrinking torus, 85, 88 rescaled, 2, 4, 13, 40, 62, 68, Simons’ identity, 113 71, 72, 88, 94, 103, 151 singular point, 81 self-similar, 2 singular set, 81, 85, 86 smooth, 3, 20 dimension of, 2, 3, 85 smooth extension of, 46 singularity, 23, 28, 29, 81, 87 space-time track of, 84, 146 formation in finite time, 23, special, 3, 7 28, 29 translating, 16, 31 smooth extension of solution, 46 unique, 16 smooth solution, 3, 20 weak, 49 smoothness estimate, 39, 90, 132, space-time track, 60, 67, 84, 146 135 special solution, 3, 7 Sobolev inequality, 60 sphere comparison, 19, 23, 26, 27, logarithmic, 59 52 Michael–Simon, 60, 76, spiral curve, 20 139 standard estimates, 40 solution stationary limits, 9 Brakke, 4, 49, 56, 72, 84 steepest descent flow, 9 coming out of a cone, 17 subsolutions, 76 compact, 18, 19, 48 surface of revolution, 10, 88 entire graph, 3, 19, 32 generalised, 3, 72 tangent flow, 71 graph, 12, 13, 16–18, 34 tangent space, 8, 109 homothetic, 3, 4, 10, 61 approximate, 82, 102, 126 homothetically contracting, tangent vector field, 110 12, 13, 53, 68, 72, 74, tangential diffeomorphisms, 8, 11, 151 14, 15, 72 INDEX 165 tangential divergence, 25, 115 tangential gradient, 25, 110, 115 tangential projection, 8, 114 test function, 50 spherically shrinking, 51, 91 time-dependent, 50, 51 Thurston’s geometrisation conjecture, 1 tilt-excess, 131 torus, 10 total energy, 89, 98 total time derivative, 25 translating solution, 16, 31 type-I condition, 72 uniqueness of solution, 16 unit density, 87 unit density hypothesis, 82, 83, 85 unit normal field, 7, 14, 110 upper-semicontinuity of Gaussian density, 70

Vitali covering theorem, 100, 125 volume of unit n-ball, 82 weak solution, 49 Weingarten map, 111 White’s local regularity theorem, 4, 90, 102, 132 White’s stratiﬁcation theory, 90

Young’s inequality, 34, 131 Progress in Nonlinear Differential Equations and Their Applications

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PNLDE 32 Geometrical Optics and Related Topics Ferrnccio Colombini and Nicolas Lerner, editors

PNLDE 33 Entire Solutions of Semilinear Elliptic Equations I. Kuzin and S. Pohowev

PNLDE 34 Stability of Functional Equations in Several Variables Donald H. Hyers'. George Isac and Themistocles M. Rassias PNLDE 35 Topics in Nonlinear Analysis: The Herbert Amann Anniversary Volume Joachim Escher and Cieri Simonel/, editors

PNLDE 36 Variational Problems with Concentration Manin Flucher

PNLDE 37 Implicit Partial Differential Equations Bernard Dacorogna and Paolo Marcelfini

PNLDE 38 Systems of Conservation Laws: Two-Dimensional Riemann Problems Yuxi ZJr.eng

PNLDE 39 Linear and Nonlinear Aspects of Vortices: The Ginzburg-Landau Model Frank Pacard and TriSlan Riviere

PNLDE 40 Recent Trends in Nonlinear Analysis: Festschrift Dedicated to Alfonso Vignoli on the Occasion of his Sixticth Birthday JUrgen Appell, editor

PNLDE 41 Quasi-hydrodynamic Semiconductor Equations Ansgar JUngel

PNLDE 42 Semigroups of Operators: Thcory and Applications A. V. Balakrishnan, editor

PNLDE 43 Nonlinear Analysis and ilS Applications to Differential Equations M.R. Grosshino, M. Ramos, C. Rebelo, and L Sanchez. editors

PNLDE 44 The Monge-Ampere Equation Cristian E. Cwilrrel

PNLDE 45 Spati~ Patterns: Higher Order Models in Physics and Mechanics LA. Peletierand w.c. Troy

PNLDE 46 Carleman Estimates and Applications to Uniqueness and Conlrol Theory Ferrucio Colombini and Claude Zui/y, editors

PNLDE 47 Advances in the Theory of Shock Waves Heinrich Freis/Uhler and Anders Supeny, editors

PNLDE 48 Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics S.N. An/on/se!', J.r. Dlal, and S. Shmarev

PNLDE 49 Variational and Topological Methods in the Study of Nonlinear Phenomena v. Bend, C. Cerami, M. Degiovonni, D. Fortunato, F. Gianl1lJni, and A. M. Micheletti, editors PNLDE 50 Topics in Parabolic Differential Equations A. Lo,-,nzi and B. Rut

PNlDE 51 Varialional Methods ror Discontinuous StruetW'eS G. dd Maso and F. Tomartlli, editors

PNlDE.52 Partial Differential Equations and Mathematical Physics: In Memory or Jean leray K. Kajilani and J. \1:lillanl, edilOrs

PNlDE.53 flow Unes and Algebraic Invariants in Cont3C1 Form Geometry A. Bohri

PNDlE 54 Nonlinur Models and Melhods D. Lupo, C Pagani, and B. Ruff

PNLDE.5.5 Evolution Equations: Applications to Physics, Industry, Ure Science and Economics G. Lumer ondM. Iannelli

PNlDE 56 A Stability Technique ror Evolution Panial Differential Equ3lions: A Dynamical Systems Approach Victor A. Galaktionov and Juan Luis V6zqlU'z

PNLDE.57 Regularity Theory for Mean Curvolu,-, Flow Klaus Eder