The Mean Curvature Flow in Minkowski Spaces

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Euclidean metric without quadratic restriction. Curvature flows in Minkowski spaces were investigated by other authors, see e.g. [1, 2,22]. Note that the curvature flows in [1, 2,22] are called “anisotropic curvature flows”. In fact, they can be thought as curvature flows on the Minkowski spaces. To the best of our knowledge, there are less works on higher dimensional mean curvature flow concerning about detailed convergence on Finsler manifolds. Compared with the case of Riemannian geometry, in our case, we need to overcome some obstructions, the major one is how to get a priori estimate from the PDE point of view. Due to a lot of non-Riemannian geometric quantities, most of classical approach to prove a priori estimates by Huisken fails. Particularly, when we write the flow function as a scalar function of the graph function r over the inverse Minkowski hypersphere, the evolution equation for ∂tr behaves not well. Also, the evolution equation for the anisotropic second fundamental form is quite bad. In this paper, our motivation is to better understand the mean curvature flow on Finsler manifolds, hoping that in further studies we can detect interesting geometric objects by running the mean curvature flow or similar flows to time infinity without developing any singularity or after handling possible singularities. As a first step, we will give a definition of the mean curvature flow on Finsler manifolds. It is not an easy way to define mean curvature flow of mutually compatible fundamental geometric structures on Finsler manifolds. Our mean curvature flow equation has the following form:

∂tϕ(p,t)= Hˆ (p,t)n(p,t), ( ϕ(p, 0) = ϕ0(p), where Hˆ (p,t) and n(p,t) are respectively the anisotropic mean curvature and the inner pointing unit normal of the hypersurface ϕt at the point p M. For the exact definition ∈ of Hˆ we refer to Section 2.2.1. We also provide some examples of the mean curvature flow in Finsler geometry. Then, due to the complexity of the evolution equation, we focus on the mean curvature flow on the Minkowski spaces in the rest of the paper and obtain some interesting results. The contents of this paper are organized as follows. In Section 2, we provide some fundamental concepts and formulas which are necessary for the present paper. In Section 3, we give the definition of the mean curvature flow on Finsler manifolds. In Section 4, we give some examples of the mean curvature flow in Finsler geometry. In Section 5, for Minkowski spaces we use a strictly parabolic second-order quasi-linear partial differential equation and prove the existence and uniqueness of solutions of mean curvature flow. In Section 6, we give some evolution equations. As some applications, In Section 7 and Section 8, we provide the convexity, mean convexity and comparison principles for the mean curvature flow on the Minkowski spaces.

2 Preliminaries

2.1 Finsler manifolds

Here and from now on, we will use the following convention of index ranges unless other stated: 1 i, j, k, n; 1 a, b, c, n 1; 1 α, β, γ, n+1; 1 I, J, 2n ≤ ···≤ ≤ ···≤ − ≤ ··· ≤ ≤ ···≤ 3 throughout this paper. For simplicity, from now on we will follow the summation conven- ∂u ∂2u tion and frequently use the notations F = F (y), Fi = Fyi (y), ui = ∂xi , uij = ∂xi∂xj and so on. We assume that M is an n-dimensional oriented smooth manifold. Let T M be the tangent bundle over M with local coordinates (x,y), where x = (x1, ,xn) and y = · · · (y1, ,yn). A Finsler metric on M is a function F : T M [0, ) satisfying the · · · −→ ∞ following properties: (i) F is smooth on T M 0 ; (ii) F (x, λy) = λF (x,y) for all λ > 0; \{ } (iii) the induced quadratic form g is positive-deﬁnite, where

i j 1 2 g := gij(x,y)dx dx , gij(x,y)= [F ] i j . ⊗ 2 y y The projection π : T M M gives rise to the pull-back bundle π∗T M over T M 0 . As −→ \{ }j is well known, on π∗T M there exists uniquely the Chern connection with ∂ = ω ∂ ∇ ∇ ∂xi i ∂xj satisfying ωi dxj = 0, j ∧ k k k k 1 k j k dgij gikω gkjω = 2FCijkδy , δy := (dy + y ω ), − j − i F j 1 ∂gij where Cijk = 2 ∂yk is called the Cartan tensor. i ∂ i ∂ Let X = X ∂xi be a vector field. Then the covariant derivative of X along v = v ∂xi with respect to w TxM 0 is defined by ∈ \{ } ∂Xi ∂ DwX(x) := vj (x)+Γi (w)vj Xk(x) , v ∂xj jk ∂xi i i k where Γjk = ωj(∂x ). Let : T M T ∗M denote the Legendre transformation, which satisfies (0) = 0 L −→ L and (λy)= λ (y) for all λ> 0 and y T M 0 . Then : T M 0 T ∗M 0 is L L ∈ \ { } L \ { } −→ \ { } a norm-preserving C∞ diffeomorphism. For a smooth function f : M R, the gradient −1 −→ vector of f at x M is defined as f(x) := (df(x)) TxM, which can be written as ∈ ∇ L ∈ gij(x, f) ∂f ∂ , df(x) = 0, f(x) := ∇ ∂xj ∂xi 6 ∇ ( 0, df(x) = 0.

2 ∗ Set Mf := x M df(x) = 0 . We deﬁne f(x) T M TxM for x Mf by using the { ∈ | 6 } ∇ ∈ x ⊗ ∈ following covariant derivative

2 ∇f f(v) := D f(x) TxM, v TxM. ∇ v ∇ ∈ ∈ Let dµ = σ(x)dx1 dxn be an arbitrary volume form on (M, F ). The divergence ∧···∧ i ∂ of a smooth vector field V = V ∂xi on M with respect to dµ is defined by 1 ∂ ∂V i ∂ ln σ divV := σV i = + V i . σ ∂xi ∂xi ∂xi Then the Finsler-Laplacian of f can be defined by

1 ∂ ij ∆f := div( f)= σg ( f)fj . ∇ σ ∂xi ∇ 4 The Mean Curvature Flow in Minkowski Spaces

2.2 Anisotropic hypersurfaces

Now let ϕ : M n (N n+1, F ) be an embedded hypersurface of Finsler manifold (N, F ), → where M is an n-dimensional oriented smooth manifold. For simplicity, we will use (x,y) ∈ T M, where x = ϕ(u), y = dϕv for (u, v) T M. Set ∈ (M)= ξ ϕ−1T ∗N ξ(dϕ(X)) = 0, X Γ(T M) , V { ∈ | ∀ ∈ } which is called the normal bundle of M in (N, F ). There exist exactly two unit normal vector ﬁelds n on M such that n = −1( ν), where ν (M) is a unit 1-form. In ± ± L ± ∈ V general, n is not necessarily n unless F is reversible. + − − Let n = −1(ν) be a given normal vector of M in (N, F ), and putg ˆ := ϕ∗gn. Then L (M, gˆ) is a Riemannian manifold. We also call (M, gˆ) an anisotropic hypersurface of (N, F ) to distinguish it from an isometric immersion hypersurface of (N, F ).

α β gˆij = gαβ(n)ϕi ϕj , where ∂ϕα xα = ϕα(u), yα = ϕαvi, ϕα = . i i ∂ui

Similar to the Riemannian case, we have the Gauss-Codazzi equations as follows ( [14]) n DX Y = X Y + h(X,Y )n, n ∇ (2.1) DX n = AX, X,Y Γ(T M), −b ∈ where n h(X,Y ) :=g ˆ(AX,Y ) =g ˆ(n, D Y ), X,Y Γ(T M). (2.2) X ∈ is a torsion-free linear connection on M and satisfies ∇ n ( X gˆ)(Y,Z) = 2Cn(D n,Y,Z)= 2Cn(AX, Y, Z), (2.3) b ∇ X − which shows that is not the Levi-Civita connection on Riemannian manifold (M, gˆ). ∇b A : TxM TxM is linear and self-adjoint with respect tog ˆ, which is called the Weingarten → transformation (orbShape operator). So h is bilinear and h(X,Y ) = h(Y, X), which is called the anisotropic second fundamental form of M in (N, F ). Moreover, we call the eigenvalues of A, k , k , , kn , the anisotropic principal curvatures of M with respect 1 2 · · · −1 to n. If A(X) = kX, X Γ(T M), or equivalently, k = k = = kn , we call M an ∀ ∈ 1 2 · · · −1 anisotropic totally umbilic hypersurface of Finsler manifold (N, F ). Define ( [14,21]) n−1 Hˆ := Trh = ka. (2.4) a X=1 Hˆ is called the anisotropic mean curvature of M in (N, F ). We remark that when (N, F ) is a Minkowski space, Hˆ is just the anisotropic mean curvature defined in many papers like [1] and [22]. Let dµ = σ(x)dx1 dxn+1 be an arbitrary volume form on (N, F ). The induced ∧···∧ volume form on M determined by dµ can be defined by

∗ 1 n+1 dµn = σ(ϕ(u))ϕ (in(dx dx )), u M, ∧···∧ ∈ 5 where in denotes the inner multiplication with respect to n. Consider a smooth variation ϕt : M N, t ( ε, ε), such that ϕ = ϕ. From [14], we know that the ﬁrst variation of → ∈ − 0 the induced volume along the direction X. dVol(t) = dµn (X)dµn, (2.5) dt t=0 − M H Z where dµn is called the dµn-mean curvature form of ϕ. H := dµn (n) is called the H H dµn-mean curvature of M in (N, F ). From [21]( also see [14]), we know that

H = Hˆ + S(n), (2.6)

i δ √det(gij ) where S(y)= y δxi ln σ(x) , which called the S-curvature of (N,F,dµ). M is called weakly convex (mean convex, resp.) if h is nonnegative definite (Hˆ 0 resp.). We say h ≥ is strictly convex in M if h is positive definite in M. Lemma 2.2.1. ( [14]) In a Minkowski space (N, F ) with the BH(resp. HT)-volume form dµ, Hˆ = H and we have the fllowing Gauss-Codazzi equations:

gˆ(R(X,Y )Z,W )=h(Y,Z)h(X,W ) h(X,Z)h(Y,W ), − (2.7) ( X h)(Y,Z) =( Y h)(X,Z), b ∇ ∇ for any X,Y,Z,W bΓ(T M), wherebRˆ is the curvature tensor of the induced connection ∈ and ( X h)(Y,Z) := Xhˆ(Y,Z) hˆ( ˆ X Y,Z) hˆ(Y, ˆ X Z). ∇ ∇ − ∇ − ∇ b b 3 The Mean Curvature Flow in Finsler manifolds

The geometric evolution equation (1.1) is known as the mean curvature flow in Riemannian geometry. Similarly, we introduce the concept of the mean curvature flow in the Finsler setting: n n+1 n Definition 3.1. Let ϕ0 : (M , gˆ) (N , F ) be a smooth immersion of M into n+1 → a Finsler manifold (N , F ). The mean curvature flow of ϕ0 is a family of smooth n+1 immersions ϕt : M (N , F ) for t [0, T ) such that setting ϕ(p,t) = ϕt(p) the map → ∈ ϕ : M [0, T ) (N n+1, F ) is a smooth solution of the following system of PDE’s × → ∂ ϕ(p,t)= Hˆ (p,t)n(p,t), t (3.1) ( ϕ(p, 0) = ϕ0(p), where Hˆ (p,t) and n(p,t) are respectively the anisotropic mean curvature and the inner pointing unit normal of the hypersurface ϕt at the point p M. ∈ The following Gauss-Weingarten relations will be fundamental,

α j α α α α α β γ k α ∂in = h ϕ , ϕ = hijn = ϕ +Γ ϕ ϕ Γ ϕ , (3.2) − i j i|j ij βγ i j − ij k α α k where ϕi|j denotes the covariant derivative of ϕi and Γij is the Christoﬀelb symbols with respects tog ˆ. Notice that, by these relations, it follows that b ij ij ˆ ∆ϕ =g ˆ ϕi|j =g ˆ hijn = Hn. (3.3)

b 6 The Mean Curvature Flow in Minkowski Spaces

Using equation (3.3), this system can be rewritten in the appealing form

∂tϕ = ∆ϕ.

b Proposition 3.1.(Geometric Invariance under Tangential Perturbations). If a smooth n+1 family of immersions ϕt : M (N , F ) satisﬁes the system of PDE’s →

∂tϕ(p,t)= Hˆ (p,t)n(p,t)+ X(p,t), (3.4) ( ϕ(p, 0) = ϕ0(p), where X is a time-dependent smooth vector field along M such that X(p,t) belongs to dϕt(TpM) for every p M and every time t [0, T ), then, locally around any point in ∈ ∈ space and time, there exists a family of reparametrizations (smoothly timedependent) of the maps ϕt which satisfies system (3.1). If the hypersurface M is compact, one can actually find uniquely a family of global reparametrizations of the maps ϕt as above for every t 0, leaving the initial immersion ϕ unmodified ≥ 0 and satisfying system (3.1). n+1 Conversely, if a smooth family of moving hypersurfaces ϕt : M (N , F ) can be globally → reparametrized for t 0 in order that it moves by mean curvature, then the map ϕ has to ≥ satisfy the system above for some time-dependent vector field X with X(p,t) dϕt(TpM). ∈ Proof. First we assume that M is compact; we will produce a smooth global parametriza- tion of the evolving sets in order to check the definition. By the tangency hypothesis, the time-dependent vector field on M given by Y (q,t)= −1 [dϕt] (X(q,t)) is globally well defined and smooth. − Let Ψ : M [0, T ) M be a smooth family of diffeomorphisms of M with Ψ(p, 0) = p × → for every p M and ∈ ∂tΨ(p,t)= Y (Ψ(p,t),t). (3.5)

This family exists, is unique and smooth, by the existence and uniqueness theorem for ODE’s on the compact manifold M. Considering the reparametrizations ϕ(p,t)= ϕ(Ψ(p,t),t), then

e k ϕi(p,t)=ϕk(Ψ(p,t),t)Ψi , k k k ϕij(p,t)=ϕkl(Ψ(p,t),t)Ψi Ψj + ϕk(Ψ(p,t),t)Ψij, e k l gij(p,t)=gkl(Ψ(p,t),t)Ψi Ψj, e n(p,t)=n(Ψ(p,t),t), e α β γ hij(p,t)=gne(n, ϕij +Γβγϕi ϕj ) e α β γ k l =gn(n, ϕ +Γ ϕ ϕ )Ψ Ψ e kl βγ k l i j e e ke le =hkl(Ψ(p,t),t)Ψi Ψj, H(p,t)=Hˆ (Ψ(p,t),t).

e 7

From which, one has

∂tϕ(p,t)=∂tϕ(Ψ(p,t),t)+ dϕ(Ψ(p,t))(∂tΨ(p,t)) =Hˆ (Ψ(p,t),t)n(Ψ(p,t),t)+ X(Ψ(p,t),t)+ dϕ(Ψ(p,t))(Y (Ψ(p,t),t)) e =Hˆ (Ψ(p,t),t)n(Ψ(p,t),t)+ X(Ψ(p,t),t) dϕ(Ψ(p,t))([dϕ(Ψ(p,t))]−1 X(Ψ(p,t),t)) − =Hˆ (Ψ(p,t),t)n(Ψ(p,t),t) =H(p,t)n(p,t).

Hence, ϕ satisfies system (3.1) and ϕ0 = ϕ0. e e Conversely, this computation also shows that if ϕ(p,t) = ϕ(Ψ(p,t),t) satisfies system (3.1), the family of diffeomorphismse Ψ : M [0, T ) M must solve equation (3.5), hence, × → it is unique if we assume Ψ( , 0) = IdM in order that the map ϕ is unmodified. · e 0 In the noncompact case, we have to work locally in space and time, solving the above system of ODE’s in some positive interval of time in an open subset Ω M with compact ⊂ closure, then obtaining a solution of system (3.1) in a possibly smaller open subset of Ω and some interval of time. Assume now that the reparametrized map ϕ(p,t) = ϕ(Ψ(p,t),t) is a mean curvature flow. Differentiating, we get

∂tϕ(p,t)=∂tϕ(Ψ(p,t),t)+edϕ(Ψ(p,t))(∂tΨ(p,t)) =H(p,t)n(p,t) e =Hˆ (Ψ(p,t),t)n(Ψ(p,t),t) e e that is, −1 ∂tϕ(q,t)= Hˆ (q,t)n(q,t) dϕ(q)(∂tΨ(Ψ (q),t)), − t for every q M and t [0, T ). Then, the last statement of the proposition follows by ∈ ∈ −1 setting X(q,t)= dϕt(q)(∂tΨ(Ψ (q),t)). − t ⊔⊓ Corollary 3.1. If a smooth family of hypersurfaces ϕt = ϕ( ,t) satisfies gn(∂tϕ, n)= Hˆ , · then it can be everywhere locally reparametrized to a mean curvature flow. If M is compact, this can be done uniquely by global reparametrizations, without modifying ϕ0. We give a more geometric, alternative definition of the mean curvature flow as follow. n+1 Definition 3.2. We still say that a family of smooth immersions ϕt : M (N , F ), → for t [0, T ), a mean curvature flow if locally at every point, in space and time, there ∈ exists a family of reparametrizations which satisfies system (3.1). n+1 Let A(t) is the area of closed hypersurface ϕt : M (N , F ) with respect to dµn. → Using (2.6) and (2.5), during the flow (3.1), we have

dA(t) 2 = (Hˆ + HSˆ (n))dµn. (3.6) dt − ZM Proposition 3.2. If a smooth family of closed hypersurfaces ϕt = ϕ( ,t) satisﬁes (3.1) · and the Finsler manifold (N n+1,F,dµ) has vanishing S-curvature, then the area of the hypersurfaces is nonincreasing i.e.,

dA(t) 2 = Hˆ dµn 0. dt − ≤ ZM 8 The Mean Curvature Flow in Minkowski Spaces

And thus we have the estimate

Tmax 2 dt Hˆ dµn A(0). ≤ Z0 ZM

4 Examples

Example 4.1. (Motion of Level Sets).

n+1 Level set representation Mt = f(ϕ(p,t),t) = 0 , where f : N [0, T ) R, p M. { } × → ∈ Then we have f f n = ∇ ,H = div ∇ = Hˆ + S(n). F ( f) − F ( f) ∇ ∇ We compute d(f(ϕ(p,t),t)) 0= = g f ( f, ∂tϕ)+ ∂tf. dt ∇ ∇

If ϕt satisﬁes system (3.1), then g f ( f, ∂tϕ)= F ( f)Hˆ and thus f satisﬁes ∇ ∇ ∇ 2f( f, f) ∂tf =∆f ∇ ∇ ∇ + S( f). (4.1) − F ( f)2 ∇ ∇ Conversely, if we have a smooth function f satisfying the above equation (4.1), by Corollary 3.1, every regular level set of f( ,t) is a hypersurface moving by mean curvature. · Example 4.2. (Isoparametric hypersurfaces).

Let f be an isoparametric function on a Finsler manifold (N n+1,F,dµ) with constant S-curvature (n + 1)cF , that is, there is a smooth function a(t) and a continuous function b(t) deﬁned on J = f(Mf ) such that F ( f)= a(f), ∇ (4.2) ( ∆f = b(f). n+1 The isoparametric family of f is f = s . Let ϕt : M (N , F ) be a family of smooth { } → immersions satisfying f(ϕ(p,t)) = s(t). Then from [14], the function f˜(x,t)= f(x) s(t) satisﬁes − ′ ∂tf˜ = s (t), − 2f˜( f,˜ f˜) 2f( f, f) ∆f˜ ∇ ∇ ∇ + S( f˜) =∆f ∇ ∇ ∇ + c(n + 1)F ( f˜) − F ( f˜)2 ∇ − F ( f)2 ∇ ∇ ∇ =b(s) a(s)a′(s)+ c(n + 1)a(s) − =a(s)H.ˆ

1 If Hˆ = 0, we can set t = ′ ds. Then f˜ satisﬁes (4.1) and thus the 6 a(s)a (s)−b(s)−c(n+1)a(s) isoparametric family f = s(t) is a mean curvature ﬂow on Finsler manifolds. [14], [15] R and [23] have given a great many examples of isoparametric hypersurfaces in some Finsler space forms like Minkowski space with BH-volume (resp. HT -volume) form, Funk space and Randers sphere with BH-volume form. ⊔⊓ 9

Example 4.3. (Homothetically shrinking ﬂows in a Minkowski space).

A family hypersurfaces that simply move by contraction during the evolution by mean curvature is called a homothetically shrinking flow. From [14], we know that a given hyperplane, a isoparametric family of Minkowski hyperspheres(also called Wulff shapes) or F ∗-Minkowski cylinders are all special examples of homothetically shrinking flows in a Minkowski space (N n+1, F ). Proposition 4.1. If an initial hypersurface ϕ : M (N n+1, F ) satisfies Hˆ (p) + 0 → λgn(ϕ (p) x , n (p)) = 0 at every point p M for some constant λ> 0 and x N n+1, 0 − 0 0 ∈ 0 ∈ then it generates a homothetically shrinking mean curvature flow. Conversely, if ϕ : M [0, T ) (N n+1, F ) is a homothetically shrinking mean curvature × → flow around some point x N n+1 in a maximal time interval, then either Hˆ is identically 0 ∈ zero or gn(ϕ(p,t) x , n(p,t)) Hˆ (p,t)+ − 0 = 0, 2(T t) − for every point p M and time t [0, T ). ∈ ∈ Proof. If the condition is satisfied, we consider the homothetically shrinking flow ϕ(p,t)= x + √1 2λt(ϕ (p) x ). 0 − 0 − 0 It is easy to prove that Hˆ (p, 0) n(p,t)= n0(p), Hˆ (p,t)= . √1 2λt − Then we see that

λgn(ϕ0(p) x0, n(p,t)) Hˆ (p, 0) gn(∂tϕ(p,t), n(p,t)) = − = = Hˆ (p,t). − √1 2λt √1 2λt − − Hence, by Corollary 3.1, this is a mean curvature flow of the initial hypersurface ϕ0, according to Definition 3.2. Conversely, if the homothetically shrinking evolution ϕ(p,t)= x + f(t)(ϕ (p) x ) 0 0 − 0 is a mean curvature flow, for some positive smooth function f : [0, T ) R with f(0) = 1, ′ → limt T f(t)=0 and f (t) 0, by Corollary 3.1 we have gn(∂tϕ, n)= Hˆ , hence → ≤ Hˆ (p,t)=gn(∂tϕ(p,t), n(p,t)) ′ =f (t)gn(ϕ0(p) x0, n(p,t)) ′ − (4.3) f (t) = gn(ϕ(p,t) x , n(p,t)) f(t) − 0 and Hˆ (p, 0) =f(t)Hˆ (p,t) ′ (4.4) =f(t)f (t)gn0 (ϕ0(p) x0, n0(p)), − ′ as n(p,t) = n (p). If Hˆ = 0 at some point, we have that f(t)f (t) is equal to some 0 6 constant C for every t [0, T ), combining (4.3) and (4.4). Hence, f(t) = √2Ct +1 as ∈ t f(0) = 1 and since limt T f(t) = 0, we conclude f(t)= 1 . The thesis then follows → − T from (4.3). q ⊔⊓ 10 The Mean Curvature Flow in Minkowski Spaces

5 Short Time Existence of the Flow

From this section, we suppose that (N n+1, F ) is an n + 1-dimensional Minkowski space. Theorem 5.1. For any initial, smooth and compact hypersurface of N n+1 given by an immersion ϕ : M (N n+1, F ), there exists a unique, smooth solution of system (3.1) 0 → in some positive time interval. Moreover, the solution continuously depends on the initial ∞ immersion ϕ0 in C . Proof. Let ϕ : M (N n+1, F ) be a smooth immersion of a compact n-dimensional 0 → manifold. For the moment we assume that this hypersurface is embedded, hence the inner pointing unit normal vector field n0 (about Euclidean metric)is globally defined and smooth. We look for a smooth solution ϕ : M [0, T ) (N n+1, F ) of the parabolic problem × → (3.1) for some T > 0. Since we are interested in a solution for short time, we can forget about the immersion condition (dϕt nonsingular) as it will follow automatically by the smoothness of the solution and by the fact that ϕ0 is a compact immersion, when t is close to zero. Keeping in mind Proposition 3.1 and Corollary 3.1, if we find a smooth solution of the problem

gn(∂ ϕ(p,t), n(p,t)) = H(p,t), t (5.1) ( ϕ(p, 0) = ϕ0(p), then we are done. We consider the regular tubular neighborhood Ω = x N n+1 d(x, ϕ (M)) < ε , { ∈ | 0 } which exists for ε> 0 small enough. By regular we mean that the map Ψ : M ( ε, ε) Ω × − → defined as Ψ(p,s)= ϕ0(p)+sn0(p) is a diffeomorphism, where n0 is the unit normal vector n+1 of ϕ0 with respect to the Euclidian metric in N . Any small deformation of ϕ0(M) inside Ω can be represented as the graph of a height function f over ϕ0(M) and conversely, to any small function f : M R we can associate the hypersurface M f Ω given by → ⊂ ϕ(p) = ϕ0(p)+ f(p)n0(p). We want to compute now the equation for a smooth function f, time-dependent, in order that ϕ satisfies system (5.1). Obviously, as f( , 0) gives the · hypersurface ϕ , we have f(p, 0) = 0 for every p M. 0 ∈ First we compute the metric and the normal of the perturbed hypersurfaces. It is easy to see that k ϕi(p, 0) =ϕ i(p)+ n (p)fi(p,t) f(p,t)h (p, 0)ϕ k(p, 0) 0 0 − i 0 (5.2) =ϕi(p,f,∂f), where ∂f = (f1,...,fn). Then the induced metric and normal vector of ϕ with respect to the Euclidian metric , in N n+1 are h i

g (p,t)= ϕi(p,t), ϕj (p,t) ij h i 2 kl =g (p, 0) 2f(p,t)hij(p, 0) + f (p,t)hikg hlj(p, 0) + fi(p,t)fj(p,t) ij − =gij(p,f,∂f), n =n dϕ f, 0 − ∇ 11 where f is the gradient vector with respect to g. Hence we know that ∇ n n dϕ f ν = = 0 − ∇ = ν(p,f,∂f), F ∗(n) F ∗(n dϕ f) 0 − ∇ n = −1(ν)= n(p,f,∂f), L gˆij(p,t)=gn(ϕi(p,t), ϕi(p,t)) =g ˆij(p,f,∂f).

Notice that the normal, the metric and thus its inverse depend only on ﬁrst space derivatives of the function f. Moreover, as f(p, 0) = f(p, 0) = 0, everything is smooth ∇ and since M is compact, when t is small the denominator of the above expression for the normal is uniformly bounded below away from zero (actually it is close to one). From (5.2), we have

ϕij = ϕ0ij + n0fij + fn0ij + fin0j + fjn0i. Then, we ﬁnd the second fundamental form, 1 h =gn(n, ϕ )= ν(ϕ )= h ij ij ij F ∗(n) ij

n dϕ f, ϕij =h 0 − ∇ i F ∗(n dϕ f) 0 − ∇ 1 = h (p, 0) + f + f n , n dϕ f, ϕ ∗ ij ij 0 0ij 0ij F (n0 dϕ f) h i − h ∇ i − ∇ fi dϕ f, n j fj dϕ f, n i dϕ f,fn ij − h ∇ 0 i− h ∇ 0 i − h ∇ 0 i 1 = fij + Pij(p,f,∂f), F ∗(n dϕ f) 0 − ∇ where Pij is a smooth form when f and ∂f are small, hence for t small. The mean curvature is then given by ij H =ˆg hij

1 ij = gˆ fij + P (p,f,∂f), F ∗(n dϕ f) 0 − ∇ ij −1 ij where (ˆg ) = (ˆgij) = (ˆg (p,f,∂f)) and P is a smooth function, assuming that f and f are small. ∇ We are ﬁnally ready to write down the condition gn(∂tϕ(p,t), n(p,t)) = H(p,t) in terms of the function f,

(∂tf)gn(n, n0)= gn(∂tϕ(p,t), n(p,t)) = H(p,t)

1 ij = gˆ fij + P (p,f,∂f). F ∗(n dϕ f) 0 − ∇ 1 Notice that gn(n, n0)= ν(n0)= ∗ , we get F (n0−dϕ∇f)

ij P (p,f,∂f) ∂tf =g ˆ fij + ν(n0) ij =g ˆ fij + Q(p,f,∂f), 12 The Mean Curvature Flow in Minkowski Spaces where Q(p, , ) is a smooth function when its arguments are small. · · Then, if the smooth function f : M [0, T ) R solves the following partial diﬀerential × → equation (before we had to deal with a system of PDE’s)

∂ f =g ˆijf + Q(p,f,∂f), t ij (5.3) ( f(p, 0) = 0, then ϕ(p,t) = ϕ0(p)+ f(p,t)n0(p) is a solution of system (3.1) for the initial compact embedding ϕ0, in a positive time interval.

Conversely, if we have a mean curvature flow ϕ of ϕ0, for small time the hypersurfaces ϕt are embedded in the tubular neighborhood Ω of ϕ0(M), then the function f(p,t) = −1 −1 π(−ε,ε)[Ψ (ϕ(p,t))] is smooth and f(p, 0) = π(−ε,ε)[Ψ (ϕ(p, 0))] = π(−ε,ε)[(p, 0)] = 0, where π is the projection map on the second factor of M ( ε, ε), hence, by the (−ε,ε) × − above computations f must solve problem (5.3). PDE (5.3) is a quasilinear strictly parabolic equation. In particular it is not degenerate (in some sense, passing to the height function f we killed the degeneracy of systems (3.1) and (5.1)) hence, we can apply the (almost standard) theory of quasilinear parabolic PDE’s. The proof of a general theorem about existence, uniqueness and continuous dependence of a solution for a class of problems including (5.3) can be found again in [17] (see Appendix A in [17]). Using the unique solution f of problem (5.3) we consider the associated map ϕ(p) = ϕ0(p)+f(p)n0(p), we possibly restrict the time interval in order that ϕt are all immersions and we apply Corollary 3.1 to reparametrize globally the hypersurfaces in a unique way in order to get a solution of system (3.1). This association is one-to-one, as long as one remains inside the regular tubular neighborhood Ω, hence, existence, uniqueness, smoothness and dependence on the initial datum of a solution of system (3.1) follows from the analogous result for quasilinear parabolic PDE’s. If the hypersurface is not embedded, that is, it has self-intersections, since locally every immersion is an embedding, we only need a little bit more care in the definition of the height function associated to a mean curvature flow (a regular global tubular neighborhood is missing), in order to see that the correspondence between a map ϕ and its height function f is still a bijection, then the same argument gives the conclusion also in the nonembedded case. ⊔⊓

6 Evolution of Geometric Quantities

Proof. Using (2.7), we calculate the second covariant derivative of H and obtain

Hence, kl kl kl kl gˆ h = H gˆ hkl gˆ h gˆ h . kl|ij |ij − |ij − |i kl|j − |j kl|i We calculate the Laplacian of hij and obtain

kl kl ∆hij =ˆg hij|kl =g ˆ hik|jl kl ˆ m ˆ m =ˆg (hik lj + Rljkmh + Rljimh ) b | i k kl ˆ m ˆ m =ˆg (hkl|ij + Rljkmhi + Rljimhk ) kl kl m kl m =ˆg h +g ˆ h (hlkhjm hlmhjk) +g ˆ h (hmj hli hjihlm) kl|ij i − k − kl m 2 =ˆg h + Hh hjm A hij kl|ij i − | |gˆ m 2 kl kl kl =Hh hjm A hij + H gˆ hkl gˆ h gˆ h . i − | |gˆ |ij − |ij − |i kl|j − |j kl|i Hence, m 2 kl kl kl H = ∆hij Hh hjm + A hij +g ˆ hkl +g ˆ h +g ˆ h . (6.1) |ij − i | |gˆ |ij |i kl|j |j kl|i We calculate the ﬁrstb and second covariant derivative ofg ˆ by using (2.3) and obtain

α β γ s s gˆ = 2Cαβγ ϕ ϕ ϕ h = 2Cklsh (6.2) kl|i − k l s i − i and b α β γ q gˆ =( 2Cαβγ ϕ ϕ ϕ h ) ij|mn − i j q m |n α β γ τ q p α β γ q =2Cαβγτ ϕ ϕ ϕ ϕ h h 2Cαβγ ϕ ϕ ϕ h (6.3) i j q p m n − i j q m|n p q q =2Cijpqh h 2Cijqh . m n − m|n It follows from (6.2) and (6.3)b that b

kl ks pl α β γ m ks pl m ks pl gˆ = gˆ gˆ gˆ = 2Cαβγ ϕ ϕ ϕ h gˆ gˆ = 2Cspmh gˆ gˆ . |i − sp|i s p m i i kl α β γ m ks pl b gˆ|ij =(2Cαβγ ϕs ϕp ϕmhi gˆ gˆ )|j ks pl α β γ m ks pl α β γ m =2ˆg|j gˆ Cαβγϕs ϕp ϕmhi + 2ˆg gˆ|j Cαβγϕs ϕp ϕmhi α β γ τ q m ks pl α β γ m ks pl 2Cαβγτ ϕ ϕ ϕ ϕ h h gˆ gˆ + 2Cαβγ ϕ ϕ ϕ h gˆ gˆ − s p m q j i s p m i|j ks pl ks pl m q m ks pl m ks pl =2(ˆg gˆ +g ˆ gˆ )Cspmh 2Cspmqh h gˆ gˆ + 2Cspmh gˆ gˆ . |j |j i − j i i|j The fourth term of (6.1) can be writtenb as b b

kl α β γ m ks pl gˆ|ijhkl =(2Cαβγ ϕs ϕp ϕmhi gˆ gˆ )|jhkl ks pl α β γ m ks pl α β γ m =2ˆg|j gˆ Cαβγϕs ϕp ϕmhi hkl + 2ˆg gˆ|j Cαβγϕs ϕp ϕmhi hkl α β γ τ q m ks pl α β γ m ks pl 2Cαβγτ ϕ ϕ ϕ ϕ h h gˆ gˆ hkl + 2Cαβγ ϕ ϕ ϕ h gˆ gˆ hkl. − s p m q j i s p m i|j 14 The Mean Curvature Flow in Minkowski Spaces

The ﬁfth term of (6.1) can be written as

(1) ∂tn = dϕ H; − ∇ (2) ∂tgˆij = 2Hhij 2C(∂i, ∂j, H); − b − ∇ ij is j lj i k (3) ∂tgˆ = 2ˆg h H + 2ˆg C H ; s b lk b 2 (4) ∂tH = ∆H + H A + 2trgˆC( , A, H); | |gˆ b · ∇ m 2 k m p m p (5) ∂thij = ∆hij 2Hh hjm + A hij + 2C (h h + h h ) b − i b| |gˆ b pm i k|j j k|i + 2Ck hm hl 2Ck hmhphq . b lm i|j k − mpq i j k b

b b α j α Proof. (1) Taking derivative of ∂tϕ = Hn with respect to x and using n = h ϕ , i − i j we have k (∂tϕ)j = Hjn Hh ϕk. (6.5) − j α β Taking derivative of n gαβ(n)ϕi = 0 with respect to t, we have α β γ α β α β (∂tn )gαβ(n)ϕi + 2Cαβγ (∂tn )n ϕi + n gαβ(n)∂tϕi = 0. α β γ From (6.5) and Cαβγ n ϕi ∂tn = 0, we have α β α β (∂tn )gαβ(n)ϕ = n gαβ(n)∂tϕ i − i α β k β = n gαβ(n)(n Hi Hh ϕ ) − − i k = Hi. − α ij α Using the above formulas, we have ∂tn = gˆ ϕ Hi. Hence, − j

∂tn = dϕ H. − ∇ α β (2) Taking derivative ofg ˆij = gαβ(n)ϕi ϕj withb respect to t, we have

α β α β γ ∂tgˆij =2(∂tϕi )gαβ(n)ϕj + 2ϕi Cαβγϕj (∂tn ) β α k α α β kl γ =2gαβ(n)ϕ (n Hi Hh ϕ ) 2ϕ Cαβγϕ gˆ ϕ Hk j − i k − i j l kl = 2Hhij 2Cijlgˆ Hk. − − b 15

sj j (3) Diﬀerentiating the formulag ˆisgˆ = δi we get

ij is lj ∂tgˆ = gˆ ∂tgˆslgˆ − is lj = gˆ gˆ [ 2Hhsl 2C(∂s, ∂l, H)] − − − ∇ is j is lj =2ˆg h H + 2ˆg gˆ C(∂s, ∂l, H). s b ∇ b

(4) We calculate the evolution of H and obtainb b

ij ij ij ∂tH =∂t(ˆg hij) = (∂tgˆ )hij +g ˆ ∂thij is j is lj ij 2 kl =hij[2ˆg h H + 2ˆg gˆ C(∂s, ∂l, H)] +g ˆ [ H Hhikgˆ hlj] s ∇ ∇ij − 2 =H A + ∆H + 2trgCn( , A, H). | |gˆ ˆ b · ∇ b b

α k α b α b b (5) From ϕij = Γijϕk + hijn , we have hij = gn(n, ϕij ). Using (6.5), we have

k k k k (∂tϕb)ij = nHij Hih ϕk Hjh ϕk H∂jh ϕk Hh ϕkj. − j − i − i − i

Hence, ∂thij =∂tgn(n, ϕij) β γ α =gn(∂tn, ϕij)+ gn(n, ∂tϕij) + 2Cαβγ n (∂tn )ϕij k = gn(dϕ H, ϕij )+ Hij Hhkih − ∇ − j k k = gn(Hlϕl, Γ ϕk + hijn)+ Hij Hhkih − b ij − j k k = HlΓ gˆkl + Hij Hhkih − ij b − j kl =H|ij Hhikgˆ hlj. −b ⊔⊓

7 Consequences of Evolution Equations

The main tool in order to obtain priori estimates is the maximum principle, in particular in the context of mean curvature ﬂow. As in [1, 10, 17], we have the following maximum principles: Lemma 7.1.( [17]) Assume that g(t), for t [0, T ), is a family of Riemannian metrics ∈ on a manifold M, with a possible boundary ∂M, such that the dependence on t is smooth. Let u : M [0, T ) R be a smooth function satisfying × →

∂tu ∆ u + X(p, u, u, t), u + b(u) ≥ g(t) h ∇ ∇ ig(t) where X and b are respectively a continuous vector ﬁeld and a locally Lipschitz function in their arguments. Then, suppose that for every t [0, T ) there exists a value δ > 0 and a compact subset ∈ ′ ′ K M ∂M such that at every time t (t δ, t + δ) [0, T ] the maximum of u( ,t ) is ⊂ \ ∈ − ∩ · attained at least at one point of K (this is clearly true if M is compact without boundary). 16 The Mean Curvature Flow in Minkowski Spaces

Setting umin(t) = minp∈M u(p,t) we have that the function umin is locally Lipschitz, hence diﬀerentiable at almost every time t [0, T ) and at every diﬀerentiability time, ∈ du (t) min b(u (t)). dt ≥ min ′ As a consequence, if h : [0, T ) R is a solution of the ODE → ′ h (t)= b(h(t)),

( h(0) = umin(0),

′ ′ for T T , then u h in M [0, T ). ≤ ≥ × ′ Moreover, if M is connected and at some time τ (0, T ) we have u (τ) = h(τ), ∈ min then u = h in M [0,τ], that is, u( ,t) is constant in space. × · Lemma 7.2.( [1]) Let f 0 be bounded and C2 on A Rn (0, T ), such that ≥ ⊂ × ∂f ij i α fij + β fi + δf, ∂t ≥ where αij is non-negative definite and C2, βi is C1, and δ is bounded. Suppose there exists (x ,t ) in the interior of A such that f(x ,t ) = 0. Then f(x,t) = 0 for every (x,t) A for 0 0 0 0 ∈ which there exists a piecewise smooth path γ : [0, 1] A with γ(0) = (x ,t ), γ(1) = (x,t), → 0 0 and N ′ ij i ij γ (s)= ωi(s)α ∂i + r(s) β ∂jα (∂i ∂t) γ(s) γ(s) − γ(s) − i,j=1 X with r(s) 0 and ωi(s) R for each s [0, 1]. ≥ ∈ ∈ Lemma 7.3.( [10]) Let f(t) = sup g(t,y) : y Y , where Y is a compact set. Then { ∈ } d ∂ f(t) sup g(t,y) : y Y (t) , dt ≤ ∂t ∈ where Y (t)= y : g(t,y)= f(t) . { } Let us see some consequences of application of these maximum principles to evolution equations for curvature. A hypersurface is mean convex if H 0 everywhere. First we ≥ will show that this property is preserved by the mean curvature flow. Theorem 7.1.(Preserving mean convexity) Assume that the initial, compact hypersurface satisfies H 0. Then, under the mean curvature flow, the minimum of H is increasing, ≥ hence H is positive for every positive time. Moreover,

1 − 2 2 2 Hmin(t) Hmin(0) 1 H (0)t , ≥ − n min which gives an upper bound on the maximal existence time when Hmin(0) = 0: 6 n −2 Tmax H (0). ≤ 2 min

Proof. Arguing by contradiction, suppose that in an interval (t ,t ) R+ we have 0 1 ⊂ Hmin(t) < 0 and Hmin(t )=0(Hmin is obviously continuous in time and Hmin(0) 0). 0 ≥ 17

Let A 2 C in such an interval. Then | |gˆ ≤ 2 ∂tH = ∆H + 2trgCn(dϕ, dϕ A, dϕ H)+ H A ˆ ◦ ∇ | |gˆ implies b b ∂tHmin CHmin ≥ for almost every t (t ,t ). ∈ 0 1 C(t−s) Integrating this diﬀerential inequality in [s,t] (t0,t1) we get Hmin(t) e Hmin(s), + ⊂ ≥ then sending s t we conclude Hmin(t) 0 for every t (t ,t ) which is a contradiction. → 0 ≥ ∈ 0 1 Since then H 0 we get ≥ 2 ∂tH =∆H + 2trgCn(dϕ, dϕ A, dϕ H)+ H A ˆ ◦ ∇ | |gˆ H3 ∆b H + 2trgˆCn(dϕ, dϕ A, dϕ b H)+ . ≥ ◦ ∇ n

From Lemma 7.1, in Mt, Hb( ,t) ϕ(t), where ϕ(t) satisfyingb the ODE: · ≥ ∂ϕ 1 3 ∂t = n ϕ , ( ϕ(0) = Hmin(0).

Clearly, the solution of the ODE is

1 − 2 2 2 ϕ(t)= Hmin(0) 1 H (0)t . − n min

Then, if Hmin(0) = 0 the ODE solution ϕ(t) is always zero; so if at some positive time Hmin(τ) = 0, we have, from Lemma 7.1, that H( ,τ) is constant equal to zero on M, · but there are no compact hypersurfaces with zero mean curvature. Hence, Hmin is always increasing during the ﬂow and H is positive on all M at every positive time. By the maximum principle,

1 − 2 2 2 Hmin(t) ϕ(t)= Hmin(0) 1 H (0)t . ≥ − n min 2 2 When 1 H (0)t 0, Hmin(t) , that is to say, Mt has already blown up. − n min → → ∞ However, H is increasing and H is positive for every positive time. Hence, from 1 2 2 n −2 − H (0)T 0, we have Tmax H (0). n min ≥ ≤ 2 min ⊔⊓

Teorem 7.2.(Preserving convexity) If hij 0 at t = 0, then it remains so for t> 0, that ≥ is, if the intial, the compact hypersurface is convex, it remains convex under the mean curvature flow. Proof. It is necessary to show that h remains non-negative definite. A more general maximum principle is required here. Let Θ be defined on (p, v) T M : v = 0 , the non-zero tangent bundle, by { ∈ 6 }

h(p,t)(v, v) Θ(p,v,t)= (7.1) gˆ(p,t)(v, v) 18 The Mean Curvature Flow in Minkowski Spaces

for each p M and v TpM 0 . It suﬃces to show that Θ remains non-negative. From ∈ ∈ \{ } Proposition 6.2, the evolution equation of Θ is

1 i j ∂tΘ= (∂th Θ∂tgˆij)v v gˆ(v, v) ij − 1 m 2 k m p m p (7.2) = ∆hij 2Hh hjm + A hij + 2C (h h + h h ) gˆ(v, v) { − i | |gˆ pm i k|j j k|i k m l k m p q i j + 2Clmhb hk 2Cmpqhi h h + 2ΘHhijb + 2ΘC(∂i, ∂j, H) v v . i|j − j k ∇ } i i δ ∂ Given the coordinatesb x , v bon T M, one can observe thatb the pairb i , i forms { } { δx ∂v } a horizontal and vertical frame for T T M, where δ = ∂ Γk vj ∂ . Let dxi, δvi δxi ∂xi − ij ∂vk { } denote the local frame dual to δ , ∂ , where δvi = dvi + Γk vjdxj. Then we obtain a { δxi ∂vi } ij decomposition for T (T M 0 ) and T ∗(T M 0 ), b \{ } \{ } b T (T M 0 )= T M TM, T ∗(T M 0 )= ∗T M ∗T M, \{ } H ⊕V \{ } H ⊕V where δ ∂ T M = span , T M = span , H {δxi } V {∂vi } ∗T M = span dxi , ∗T M = span δvi . H { } V { } j ∂ i ∗ ∗ ∗ Let T = Ti ∂xj dx be an arbitrary smooth local section of π T M π T M. The ⊗ j ⊗ horizontal covariant derivative Ti|s denotes

δT j T j = i + T kΓj T jΓk . i|s δxs i ks − k is j b b The vertical covariant derivative Ti;s denotes

∂T j T j = i . i;s ∂vs

From the above deﬁnite, we have

1 gˆijΘ =ˆgij (h Θˆg ) |i|j gˆ 00|i − 00|i 00 |j 1 ij = (∆h00 2ˆg Θ|jgˆ00|i Θ∆ˆg00). gˆ00 − − b b Θ(p,v,t) = h(p,t)(v, v)/gˆ(p,t)(v, v) satisﬁes a degenerate parabolic equation on the 2n- dimensional space (p, v) T M : v = 0 : { ∈ 6 } ij i i ∂tΘ =g ˆ Θ|i|j + a Θ|i + b Θ;i + dΘ. (7.3)

The coefficients are as follows: gîj =g îj(x,t);

i i 2 ij k l mi a = a (x,v,t)= gˆ gˆ00|j + 2Clmhkgˆ gˆ00 4 ij s k i mi =− gˆ C00shj +b 2Clmhkgˆ ; gˆ00 bi = bi(x,v,t)= Hhi vj + 2Ck hp vjgˆmi +bCk hl gˆmibvj Ck hphs vjgˆmi − j pm k|j lm k |j − mps j k i j k p j mi k l t ms ip j k p s j mi = Hh v + 2C h v gˆ + 2C Cspth h gˆ gˆ v C h h v gˆ ; − j bpm k|j b lm k j b − mps j k 2 k p j bsm k b l bt sm pq j k b p q j ms d = d(x,v,t)= (2Cpmhk|jv gˆ gˆs0 + 2ClmCsqthkhjgˆ gˆ gˆp0v Cmpqhj hkv gˆ gˆs0 gˆ00 − mn p q q mn gˆ00 2 k l s mp kl + C bpqgˆ h h C qh b gˆb + A 2C C bsh h gˆ + C lgˆ Hk, 00 m n − 00 m|n 2 | |gˆ − lm 00 k p 00 i j i j where, C00s = Cijsbv v and C00pq =bCijpqv v . In particular, allb ofb these coeﬃcientsb are smooth and bounded for t [0, T ] [0, T ) on the unit sphere bundle. ∈ 0 ⊂ b b b b Therefore at the minimum point, there exists a constant C(T0) such that

∂tΘ C(T )Θ, ≥− 0 so by Lemma 7.3 (also see [10] , Lemma 3.5) the minimum of Θ decreases no faster than exponentially, and convexity is never lost. ⊔⊓

Theorem 7.3.(Preserving strict convexity) Let Mt t> be a family of convex hypersur- { } 0 faces moving by the ﬂow (3.1). Then Mt is strictly convex for each t> 0. Proof. A version of the strong maximum principle will apply to show that the region

Zt = p M : h (v, v) = 0 for some v TpM 0 { ∈ (p,t) ∈ \{ }} is open for each ﬁxed t > 0. It follows that Zt is either empty or all of M. But hij is positive deﬁnite wherever Mt touches an enclosing ball, so the only possibility is that Zt is empty.

Let p Zt and v Tp0 M such that h (v , v ) = 0. Then there exists a neigh- 0 ∈ 0 ∈ (p0,t) 0 0 borhood of p , U(p ), such that A T M, where A = U(p ) Rn. Given the coordinates 0 0 ⊂ ∼ 0 × xi,yi on A, we know that, on the 2n-dimensional space A, (7.3) can be rewritten as { } IJ I ∂tΘ= α ΘIJ +a ˜ ΘI + d˜Θ, 20 The Mean Curvature Flow in Minkowski Spaces

ij IJ gˆ ∂2Θ ∂2Θ ∂2Θ ∂Θ ∂Θ where α = ∗ ,ΘIJ = ∂xi∂xj , ∂xi∂yj , ∂yi∂yj ,ΘI = ∂xi , ∂yi and all of these ∗ ∗ ! coeﬃcients are smooth and bounded for t [0, T ] [0, T ) on the unit sphere bundle. ∈ 0 ⊂ i For every p1 U(p0), there exists a curve σ : [0, 1] U(p0) with σ(s) = (x (s)), ∈ → i i where σ(0) = p0 and σ(1) = p1. Suppose that γ(s) := (x (s), v (s)) A such that ′ i ∂ i ∂ ∈ γ (s) =x ˙ ∂xi +v ˙ ∂yi , where v(s) is a vector ﬁeld to be determined satisfying v(0) = v0. k Ik Since v0 = 0, suppose v0 = 0, we can choose ωI (s) such that ωI (s)α (σ(s)) = 0 and Ij 6 j 6 ωI (s)α (σ(s)) =x ˙ (s) for any s, where i = n + i. Let

j Ij v˙ = ωIα (7.4) ( v(0) = v0,

′ IJ k k ∂ ∂ Then γ (s) = ωIα ∂J and v (s) = v = 0, where (∂J ) = ( i , i ). Apply the strong 0 6 ∂x ∂y maximum principle Lemma 7.2 with f = Θ and r(s) identically zero on the domain A.

There exists a v = v(1) = 0 such that h 1 (v , v ) = 0. Therefore p Zt, U(p ) Zt, 1 6 (p ,t) 1 1 1 ∈ 1 ⊂ and Zt is open. ⊔⊓

8 Comparison Principle

In order to study (3.1), the following facts of convex hypersurfaces will be used. Let Sn (x ; r) := x Rn F ( (x x )) = r be the inverse Minkowski hypersphere. F− 0 { ∈ | − − 0 } For a convex hypersurface M n , we can also parametrize it as a graph over the unit inverse n Minkowski hypersphere SF− . Let X(x) π(x)= : M n Sn , (8.1) F (X(x)) → F− then we write the solution Mt to (3.1) as a radial graph X(x,t)= r(z,t)z : Sn Rn+1, (8.2) F− → −1 n α β where r(z,t)= F (X(π (z),t)), z S and gαβ( z)z z = 1. The tangent vectors and ∈ F− − the unit normal form of Mt are given by

α α α Xi = riz + rzi ,

ν β i β ν = , να = gαβ( z)( rz + r z ), F ∗(ν) − − i ∗ i ij α β where F is the dual metric of F , r = g rj and g = gαβ( z)z z . We have ij − i j α α α α α Xij = rijz + rzij + rizj + rjzi ,

α 1 α β i β hij = ναX = X gαβ( z)( rz + r z ). ij F ∗(ν) ij − − i

ij The mean curvature H =g ˆ hij can be written in terms of r(z,t) as follows:

1 ij 2 H = gˆ (r ∂ ∂ r r g 2rirj), (8.3) −F ∗(ν) ∇ i ∇ j − ij − 21

ij −1 α β where (ˆg )=(ˆgij) , gˆij = gαβ(n)Xi Xj , denotes the induced covariant derivative on n ∇ SF− . Taking derivative of (8.2) with respect to t, we have rtz = Xt = Hn. Since

i rz + r zi r H = ν(Xt)= g z rtz, − = rt, − F ∗(ν) −F ∗(ν) thus F ∗(ν) ∂ r = − H. (8.4) t r By (8.3) and (8.4), we get the evolution equation for r(z,t),

ij 2 ∂tr =g ˆ ∂ ∂ r rirj rg . (8.5) ∇ i ∇ j − r − ij If H 0, then r(z,t)= F (X(π−1(z),t)) is decreasing in [0, T ) under the mean curvature ≥ flow equation (3.1). Proposition 8.1. Assume that the initial, compact hypersurface satisfies H 0, then ≥ Mt is in fact contracting. n n n+1 Proposition 8.2. Let SF− (O1; r1) and SF− (O2; r2) be Minkowski hypersphere in M . n n If at t = 0, SF− (O1; r1) encloses the hypersurface M and SF− (O2; r2) is enclosed by the n hypersurface M, then it remains so for t > 0 and the distance between SF− (O1; r1) and n M(or SF− (O2; r2) and M) is nondecreasing in time. −z αβ αβ Proof. When r = 0, ν = r ( z). That is, n = F (−z) , g (n) = g ( z),g îj = 2 ∗ ∇ L − − r gij and F (ν)= r. Thus, from (8.5) we obtain that n ∂tr = ∆r . ∇r=0 − r

Note that the outer radius of M is rout = r(z1,t) and the inner radius of M is rin = n n r(z2,t). The distance between SF− (O1; r1) and M (or SF− (O2; r2) and M)is given by r rout ( or rin r ). we can conclude, as this analysis holds for all the pairs of points 1 − − 2 realizing the minimum, that

rout r1 r1 rout ∂t(r1 rout)= ∆(r1 r) z1 n − n − . (8.6) − − | − r1rout ≥ r1rout and r2 rin rin r2 ∂t(rin r2)= ∆(r r2) z2 n − n − . (8.7) − − | − r2rin ≥ r2rin

Then r rout ( or rin r ) is nondecreasing in [0, T ) under the mean curvature ﬂow 1 − − 2 equation (3.1). ⊔⊓

References

[1] B. Andrews, Volume preserving anisotropic mean curvature ﬂow, Indiana Univ. Math. J., 50 (2001), 783-827.

[2] B. Andrews, Motion of hypersurface by Gauss curvature, Paciﬁc J. Math. 195 (2000), 1-34. 22 The Mean Curvature Flow in Minkowski Spaces

[3] B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Diﬃeren- tial Geom., 39 (1994), 407-431.

[4] K. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 1978.

[5] B. Chow, Deforming convex hypersurfaces by the nth root of the Gausssian curvature, J. Diﬀerential Geom., 22 (1985),l 17-138.

[6] L. Evans, M. Soner and P. Souganidis, Phase transitions and generalised motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.

[7] K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math., 130 (1989), 453-471.

[8] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Diﬃerential Geom., 32 (1990), 299-314.

[9] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diﬀerential Geom., 17 (1982), 255-306.

[10] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diﬀerential Geom., 24 (1986), 153-179.

[11] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diﬃerential Geom., 20 (1984), 237-266.

[12] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84 (1986), 463-480.

[13] G. Huisken, Asymptotic behaviour for singularities of the mean curvature ﬂow, J. Diﬀerential Geom., 31 (1990), 285-299.

[14] Q. He, S. T. Yin and Y. B. Shen, Isoparametric hypersurfaces in Minkowski spaces, Diﬀerential Geom. Appl., 47 (2016), 133-158.

[15] Q. He, S. T. Yin and Y. B. Shen, Isoparametric hypersurfaces in Funk spaces, Sci. China Math. (to appear)

[16] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Diﬀerential Geom., 38 (1993), 417-461.

[17] C. Mantegazza, Lecture notes on mean curvature ﬂow, Progress in Mathematics 290 Birk¨auser/Springer Basel AG, Basel, 2011.

[18] K. Mullins, Two-dimensional motion of idealised grain boundaries, J. Appl. Phys., 27 (1956), 900-904.

[19] J. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205 (1990), 355-372.

[20] E. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math.Z., 25 (2005), 1-733. 23

[21] Z. M. Shen, Lectures on Finsler geometry, World Scientiﬁc Publishing Co., Singapore, 2001.

[22] C. Xia, Inverse Anisotropic Curvature Flow from Convex Hypersurfaces, J. Geom. Anal., (2017), DOI: 10.1007/s12220-016-9755-2.

[23] M. Xu, Isoparameteric hypersurfaces in a Randers sphere of constant ﬂag curvature, arXiv:1704.06798v1 [math.DG] 22 Apr 2017.

[24] S. T. Yin, Q. He and Y. B. Shen, On the ﬁrst eigenvalue of Finsler-Laplacian in a Finsler manifold with nonnegative weighted Ricci curvature, Sci. China Math., 57 (2014), 1057-1070.

FANQI ZENG SCHOOL OF MATHEMATICAL SCIENCE TONGJI UNIVERSITY SHANGHAI 200092 P.R. CHINA E-mail: [email protected]

QUN HE SCHOOL OF MATHEMATICAL SCIENCE TONGJI UNIVERSITY SHANGHAI 200092 P.R. CHINA E-mail: [email protected]

BIN CHEN SCHOOL OF MATHEMATICAL SCIENCE TONGJI UNIVERSITY SHANGHAI 200092 P.R. CHINA E-mail: [email protected]