Mean Curvature Flow with a Neumann Boundary Condition

on Smooth Hypersurfaces

Axel Stahl, Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72 076 Tübingen, Germany

           e-mail: axel@moebius.mathematik.uni-tuebingen.de

Tübingen, April 1995

Abstract

In this work we study the behaviour of compact, smooth, immersed manifolds with boundary, which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface Σ. We prove a very general existence result and study the formation of the singularity which occurs if Σ is a convex, umbilic hypersurface and the initial hypersurface is strictly convex.

Mathematics Subject Classification: 35K22, 53A07

1 Introduction and Main Results

The behaviour of hypersurfaces which move under various geometric flows has been studied intensively during the last years. One major topic within this research is devoted to the evolution of compact, convex hypersurfaces without boundary; here the convergence to single points in finite time has been shown for a wide class of geometric flows [13], [14], [24]. The natural question concerning the ”shape” of the developing singularity always has been of special interest. Hereby a classification with respect to the growth rate of the second fundamental form A as t approaches the singular time T_{C} is used: If there is a constant K such that |A|^{2} ≤ K(T_{C}−t) ∀t, the singularity is said to be of type I, otherwise it is called a type-II-singularity. Huisken proved [13] that convex hypersurfaces moving by mean curvature end up in a type-I-singularity, whereas it was shown by Angenent [3] that the curve shortening flow for plane curves developing a cusp will produce a type-II-singularity. Rescaling the hypersurfaces appropriately often yields a better description of what actually happens as t → T_{C}: For the mean curvature flow Huisken [13] established the convergence of the rescaled hypersurfaces to a ”round sphere” in the C^{∞}-topology. Later [15] he obtained a complete classification of the possible type-I limit surfaces with nonnegative mean curvature. Further results concerning this topic are due to Andrews [2], Hamilton [8], Stone [22], and others.

Another point of interest is the behaviour of compact surfaces with boundary which move under mean-curvature-like flows. Here in general the hypersurface is described as graph of a scalar function w : Ω × [0, T] ⟶ IR, where Ω ⊂ IR^{n} is a bounded domain, and boundary conditions are prescribed on ∂Ω × [0, T]; thus geometrically the hypersurface moves in an orthogonal cylinder. Both Dirichlet- as well as Neumann boundary conditions have been studied; results concerning the Dirichlet problem are due to Huisken [16], Stone [21], Oliker-Ural’ceva [18], and others. In the Neumann case a solution will typically converge to a hypersurface which moves by translation; the speed of the translation is determined essentially by the angle at which the hypersurface meets the boundary. This result was established in the two-dimensional case by Altschuler, Wu [4]; in the special case of an orthogonal contact angle (but in arbitrary dimensions), Huisken [16] proved the existence of a unique solution which converges to a minimal surface as t ⟶ ∞. Further results concerning a wider class of evolution equations are due to Guan [7].

Thus the evolution of compact manifolds has been studied extensively in the boundaryless case and for boundary values prescribed on orthogonal cylinders. The scope of this work is to relate these two fields by studying the boundary value problem for given Neumann data on arbitrary smooth hypersurfaces in IR^{n+1}, n ≥ 1: Establishing a sharp local gradient bound on the hypersurfaces, we prove existence and regularity results for arbitrary smooth support and initial hypersurfaces. Furthermore, in a special case, we obtain a complete classification of possible singularities.

The results presented here are essentially the results of the author’s Dissertation [23], the research upon which has been done during the years 1992-1994. The author is especially grateful to G. Huisken for his excellent tutorship during this time.

1.1 The Problem

Throughout this work, Σ denotes a smooth hypersurface in IR^{n+1}, n ≥ 1, and M^{n} a compact, smooth, orientable n-manifold with compact, smooth boundary ∂M^{n}. Furthermore, F_{0} : M^{n} ⟶ IR^{n+1} is a smooth immersion with M_{0} := F_{0}(M^{n}) and

(1.1)  \begin{aligned}∂M_{0} ≡ F_{0}(∂M^{n}) & = & M_{0} ∩ Σ, \\ ⟨ν_{0}, μ ∘ F_{0}⟩(x) & = & 0 ∀x ∈ ∂M^{n}, \end{aligned}

where ν_{0} and μ are unit normal fields to M_{0} and Σ respectively. We then define

Definition 1.1 (Mean curvature flow with Neumann boundary condition) Let F_{t} : M^{n} ⟶ IR^{n+1}, t ∈ [0, T_{C}) be a family of smooth immersions. F_{t} is said to move under the mean curvature flow with homogenious Neumann boundary condition, if

(1.2)  \{ \begin{matrix}\frac{d}{dt}F (x, t) & = & H(x, t) & ∀(x, t) ∈ M^{n} × [0, T_{C}), \\ F (., 0) & = & F_{0}, \\ F (∂M^{n}, t) & ⊂ & Σ & ∀(x, t) ∈ ∂M^{n} × [0, T_{C}), \\ ⟨ν, μ∘F ⟩(x, t) & = & 0 & ∀(x, t) ∈ ∂M^{n} × [0, T_{C}), \end{matrix}

where H denotes the mean curvature vector of the immersions.

Remark 1.2 In the sequel we assume that either Σ has no boundary or the hypersurfaces M_{t} ”keep away” from ∂Σ, i.e.

∃ε > 0 such that dist_{Σ}(F (x, t), ∂Σ) ≥ ε ∀(x, t) ∈ ∂M^{n} × [0, T_{C}),

This can be guaranteed à priori by barrier constructions, cf. section 4. Furthermore, we can often show that the solutions of equation 1.2 are contained in a compact domain K ⊂ IR^{n+1}. In many cases, this implies that K ∩ Σ is also compact, hence the second fundamental form ^{Σ}A of K ∩ Σ will be uniformly bounded. For the sake of technical simplicity we therefore require |^{Σ}A| to be uniformly bounded globally.

Given a solution F of equation 1.2, we can locally define an exterior normal field μ of Σ with respect to F: Let c : (−s, 0] ⟶ M_{t} be a smooth curve with c(0) = F (x_{0}, t) ∈ ∂M_{t}, and choose μ such that ⟨c′(0), μ∘F (x_{0}, t)⟩ ≥ 0. Thus, if Σ is the boundary of a domain G ⊂ IR^{n+1} and M_{t} ⊂ Ḡ (i.e. the surface moves in the domain), μ and the ”standard” exterior unit normal field of G will be parallel; if M_{t} ⊂ IR^{n+1}\G, they will be antiparallel. In the sequel, by ”exterior” we will always refer to a given solution as described above.

1.2 Main Results

The work is essentially divided into two parts: In sections 2 through 6 we provide a general existence result:

Theorem 1.3 Let Σ be as in Remark 1.2, and let M_{0} fulfil equations 1.1. Then there exists a unique solution to equation 1.2 on a maximal time interval [0, T_{C}). This solution is smooth for t > 0 and in the class C^{2+α,1+α/2} (with arbitrary 0 < α < 1) for t ≥ 0. Moreover, if T_{C} < ∞, then

sup{|A|^{2}(x, t) : x ∈ M^{n}} → ∞

as t → T_{C}, where A(., t) denotes the second fundamental form of the hypersurfaces M_{t}.

The crucial step proving this result is to establish a precise local gradient estimate under the assumption of bounded curvature; hereby we generalize a method of Ecker and Huisken [5]. Furthermore, in this part we discuss maximum principles and the ”C^{0}-behaviour” of hypersurfaces moving under the mean curvature flow equation 1.2.

The remainder of the work is devoted to the study of the singularity which develops as t → T_{C}, if Σ and M_{0} are (strictly) convex hypersurfaces. Under the further restriction that Σ is umbilic, i.e. the boundary of a ball B_{R}^{n+1} or a hyperplane, we can prove that convexity of the M_{t} as well as a pinching condition for the eigenvalues of the second fundamental form A are preserved under the mean curvature flow equation 1.2. Modifying an idea of Tso [24], this yields an upper bound on the growth rate of the second fundamental form of the M_{t}. Using an appropriate rescaling technique and applying results of Hamilton [11] (concerning hypersurfaces with pinched second fundamental form) and Huisken [13] (on the behaviour of compact hypersurfaces without boundary), we finally arrive at the following

Theorem 1.4 Let n ≥ 2, G = B_{R}^{n+1} (R = ∞ allowed), Σ = ∂G, and let M_{0} ⊂ G be a strictly convex, imbedded hypersurface satisfying equations 1.1. Then the hypersurfaces M_{t} of the solution to equation 1.2 shrink to a single point on Σ as t → T_{C}, ending up in a type-I-singularity. Moreover, there is a sequence t_{k} → T_{C} such that, rescaling the M_{t_{k}} appropriately, the rescaled hypersurfaces converge in the C^{∞}-topology to a hemisphere with boundary on a hyperplane.

We emphasize that the requirement of Σ being umbilic is needed exclusively to establish the conservation of the convexity and pinching condition. Consequently, the Theorem holds for arbitrary convex support hypersurfaces Σ whenever conservation of convexity and pinching can be guaranteed for the solutions of equation 1.2. Furthermore, all techniques (except for the rescaling procedure, where Hamilton’s compactness result requires n ≥ 2) are valid for plane curves. Since - as we shall see - the conservation of a positive lower bound on H already follows from the convexity of Σ, we obtain

Proposition 1.5 Let Σ be the smooth boundary of a convex domain G ⊂ IR^{2}, and let Σ have uniformly bounded curvature. Furthermore, let M_{0} ⊂ Ḡ be a convex, imbedded curve satisfying equations 1.1. Then the curves of the solution to equation 1.2 shrink to a single point on Σ as t → T_{C}.

1.3 Basic Notation and Conventions

In the sequel, g_{ij} = g_{ij}(x, t) will denote the induced metric on M^{n} with connection Γ^{k}_{ij}, A = (h_{ij}) the second fundamental form and H = g^{ij}h_{ij} the mean curvature of M_{t}.

To distinguish quantities on M^{n} and M_{t} from those on Σ, we use indices Σ,_{Σ}: Thus, g_{ij}^{Σ} is the induced metric on Σ, ^{Σ}A the second fundamental form and so on. To distinguish summations on M^{n}, M_{t} and Σ, we use the indices i, j, k... on M^{n}, M_{t} and î, ĵ, k̂... on Σ.

We will use the Einstein convention for summation of a pair of co- and contravariant indices; here the range of greek indices is from 1 to n + 1, whereas capital arabic indices range from 1 to n−1. Thus greek indices are used in IR^{n+1}, capital arabic indices on the boundaries ∂M^{n} and ∂M_{t}; the usual small arabics are reserved for calculations on M^{n}, M_{t}, Σ.

D_{α} = \frac{∂}{∂x^{α}} and ^{0}∇_{α} will denote the coordinate and covariant derivative in IR^{n+1}; ⟨., .⟩ is the scalar product in IR^{n+1}, and ∥.∥ is the norm of vectors (in IR^{n+1} as well as on M^{n}). Vector fields in IR^{n+1} and on T M^{n} will be denoted by μ, ν, ... and μ, ν, ..., respectively.

2 Short Time Existence

The first step in order to obtain an existence result for equation 1.2 is to prove that there exists a unique solution at least for a short time. This can be done by transforming the given problem into an equivalent initialboundary-value problem for a scalar function and using standard results of the parabolic theory. Ecker and Huisken have applied this method to the case of manifolds without boundary [5]; by using appropriate coordinates which account for the fact that the boundary can be curved, it is possible to adapt this technique to the given problem 1.2. As the calculations are tedious but in general straightforward, we restrict ourselves to presenting the major ideas and leave the details to the reader.

We begin by defining generalized Gaussian coordinates: Let M_{0} = F_{0}(M^{n}) be a smooth immersion of M^{n}, and let x = (x^{1}, ..., x^{n}) be coordinates on M^{n}. In a tubular neighbourhood U ⊂ IR^{n+1} of M_{0} we can define a smooth vector field ξ with the following properties:

ξ|_{M_{0}} = ν_{0} , ξ|_{Σ∩U} ∈ T Σ , ∥ξ∥ ≡ 1 .

The flux lines Φ = Φ(x, .) to the corresponding flow are perpendicular to M_{0} and tangential to Σ; hence for any given point p = Φ(x, s) ∈ U we can define x^{n+1}(p) := length of the flux line through p between p and the intersection point p_{0} = Φ(x, 0) on M_{0}, and (x^{1}, ..., x^{n})(p) := coordinates of p_{0} on M^{n}. These coordinates will be referred to as generalized Gaussian coordinates - note that with respect to these coordinates the part Σ ∩ U appears as part of an ”orthogonal cylinder”.

Now we define a scalar initial-boundary-value problem:

(2.1)  \{ \begin{matrix}\frac{∂w}{∂t}(x, t) & = & (−v H̃)(x, t) & ∀(x, t) ∈ M^{n} × [0, δ] , \\ w(., 0) & = & 0 , \\ (μ^{k} D_{k}w) & = & μ^{n+1} & ∀(x, t) ∈ ∂M^{n} × [0, δ] . \end{matrix}

Here v(x) := ⟨ν̃, ξ⟩^{−1}(x, w(x)), where ν̃ is the upper unit normal field to M̃_{t} := F̃ (M^{n}, t) := graph(w(., t)), and H̃ denotes the mean curvature of M̃_{t}. It can easily be seen that equation 2.1 is a parabolic equation with a transversal Neumann boundary condition. Hence, applying Hamilton’s results concerning the existence of solutions to linear BVPs on manifolds ([8], Chapters III and IV), we obtain a C^{∞}(M^{n}×(0, δ]) ∩ C^{0}(M^{n}×[0, δ])-solution of the linearized problem (Note that, as ⟨ν_{0}, μ∘F_{0}⟩(x) = 0 ∀x ∈ ∂M^{n}, the required compatibility condition is fulfilled). By the standard regularity theory for linear parabolic equations (e.g. [17], chapter IV) it follows that this solution actually is of the class C^{2+α,1+α/2} for any α ∈ (0, 1). Now the existence of a unique solution in C^{2+α,1+α/2}(M^{n} ×[0, δ]) as usual is a consequence of the implicit function Theorem.

Next we show that equations 1.2 and 2.1 are equivalent in the following sense: To any solution M_{t} of 1.2 there exists a unique solution M̃_{t} to 2.1 and vice versa such that M_{t} and M̃_{t} are equal as subsets of IR^{n+1}. We follow the ideas of Ecker and Huisken and define φ : M^{n} × [0, δ] ⟶ M^{n} by the following system of ordinary differential equations:

(2.2)  \{ \begin{matrix}\frac{d}{dt}φ(x, t) & = & −(D_{x}F̃)^{−1} · (\frac{∂}{∂t} F̃)^{T} (φ(x, t), t) , \\ φ(x, 0) & = & x . \end{matrix}

Here, v^{T} := v − ⟨v, ν̃⟩ν̃ denotes the tangential component of a vector. In view of the work of Ecker and Huisken [5], the only thing which remains to check is that - at least for a short time - φ is a diffeomorphism on M^{n}. To this end, it suffices to prove that φ is tangential to ∂M^{n} at the boundary, i.e.

x ∈ ∂M^{n} ⇒ φ(x, t) ∈ ∂M^{n} ∀t ∈ [0, δ].

But this is a straightforward consequence of our coordinate choice: At the boundary, we have ⟨ξ, μ⟩ = 0 which implies ⟨\frac{∂}{∂t} F̃ , μ⟩ = 0. The scalar boundary condition yields ⟨ν̃, μ⟩ = 0; combining these two relations and using equation 2.2 we obtain ⟨(D_{x}F̃) · (\frac{dφ}{dt}), μ⟩ = 0. This means that (D_{x}F̃) · (\frac{dφ}{dt}) is tangential to Σ; thus \frac{dφ}{dt} must be tangential to ∂M^{n}, for transversality is conserved by the isomorphism (D_{x}F̃). Recalling the existence of a unique solution to equation 2.1 for a short time, we have thus proved:

Theorem 2.1 (Short time existence) For any α ∈ (0, 1), equation 1.2 admits a unique solution

F ∈ C^{∞}(M^{n}×(0, δ]) ∩ C^{2+α,1+α/2}(M^{n}×[0, δ]),

where δ > 0 is a small constant. □

3 Maximum Principles

Here we provide maximum principles for scalar functions and symmetric tensors on manifolds which obey a parabolic equation with a Neumann boundary condition. The results of this section are crucial for the following and thus, despite the fact that the main ideas behind the proofs are well known, we will at least highlight the most important points.

First, let g̃ be a time independent metric on M^{n} with connection ˜∇. We then consider the following linear parabolic differential operator for scalar functions:

L(φ) := \frac{∂}{∂t}φ − a^{ij}(x, t) · ˜∇_{i} ˜∇_{j} φ − b^{i}(x, t) · ˜∇_{i} φ − c(x, t) · φ ,

with a^{ij}, b^{i}, c ∈ L^{∞}(M^{n} × [0, T]). Furthermore, let f : M^{n} × [0, T] ⟶ IR be a function with the following properties:

\begin{aligned}f(0, x, t) & ≥ & 0 , \\ |f(s, x, t) − f(0, x, t)| & ≤ & K(x, t) · |s| for |s| ≤ ρ ; \end{aligned}

here ρ > 0 is a (small) positive constant and K ∈ L^{∞}(M^{n} × [0, T]).

Finally, define a vectorfield v on ∂M^{n}×[0, T] with v ∈ T M^{n} and g̃(v, μ) > 0, where μ denotes the outward pointing unit normal field of ∂M^{n} in M^{n} (i.e. v is transversal to the boundary and outward pointing).

Theorem 3.1 (Weak maximum principle for scalar functions) Let L, f, v be as above. Let φ : M^{n} × [0, T] ⟶ IR be a continuous function which is of the class C^{2,1} in the neighbourhood of all points (x, t) ∈ M^{n} × [0, T] with |φ(x, t)| < ε (ε > 0 a small constant) and fulfils the following conditions:

(i)  φ(., 0) ≥ 0 ,
(ii) L(φ)(x, t) ≥ 0  ∀(x, t) ∈ M^{n}×(0, T] with |φ(x, t)| < ρ,
(iii) ˜∇_{v} φ(x, t) = f(φ, x, t) ∀(x, t) ∈ ∂M^{n}×(0, T] with |φ(x, t)| < ρ.

Then φ(x, t) ≥ 0 ∀(x, t) ∈ M^{n} × [0, T].

Proof: First, let φ̂ be a continuous function which is of class C^{2,1} in the neighbourhood of each point (x, t) where −ρ/10 < φ̂(x, t) < ρ. It can easily be seen that the conditions

(i) φ̂(., 0) > 0 ,
(ii) L(φ̂)(x, t) > 0 ∀(x, t) ∈ M^{n}×(0, T] with φ̂(x, t) = 0 ,
(iii) ˜∇_{v} φ̂(x, t) > 0 ∀(x, t) ∈ ∂M^{n}×(0, T] with φ̂(x, t) = 0

then imply φ̂ > 0 on M^{n} × [0, T].

Now let Λ_{δ} be a small neighbourhood of ∂M^{n} in M^{n} and define

φ̂(x, t) = \{ \begin{matrix}φ(x, t) + ε e^{αt} · Ψ(d(x)) & on Λ_{δ} , \\ φ(x, t) + ε e^{αt} · 1/2 & on M^{n}\Λ_{δ} , \end{matrix}

where

Ψ(s) := C · (s − δ)^{4} + 1/2 .

Here δ > 0, C > 0 and α > 0 are constants which will be chosen such that (for ε > 0 sufficiently small) φ̂ fulfils the above conditions: The boundary condition (iii) can be guaranteed for sufficiently small δ > 0 and sufficiently large C > 0 (depending on ∥K∥_{∞} and Δ := min_{∂M^{n}} g̃(v, μ) > 0). (ii) then follows for sufficiently large α, depending on C, δ and the data of the differential operator. The calculations are straightforward and therefore left out. Hence we have φ̂ > 0 for all ε > 0, and ε → 0 yields the result. □

Corollary 3.2 (Strong maximum principle for scalar functions) Under the assumptions of Theorem 3.1 let φ(., 0) ≢ 0 . Then

φ(x, t) > 0 on M^{n}×(0, T) ∪ M^{n}^{∘} ×{T } .

Proof: [20], chapter III. □

Next, we turn to symmetric tensors M_{ij} on M^{n} × [0, T]; here we adapt a Theorem due to Hamilton [9] to the situation of compact manifolds with boundary. To this end we let N_{ij} := p(M_{ij}, g_{ij}) be a polynomial in M_{ij} which is obtained by contracting M_{ij} with the metric g_{ij}; here, g_{ij} may depend on time. N_{ij} is said to have the zero eigenvector condition, if N_{ij}v^{i}v^{j} ≥ 0 for all zero eigenvectors of M_{ij}. We then define a parabolic operator

L(M_{ij}) := \frac{∂}{∂t}M_{ij} − ΔM_{ij} − u^{k}∇_{k} M_{ij} − N_{ij} ,

where Δ is the Laplace-Beltrami operator on M^{n} with respect to the metric g. If μ = μ(x, t) denotes the outward pointing unit normal field of ∂M^{n} in M^{n}, we obtain:

Theorem 3.3 (Weak maximum principle for symmetric tensors) Let L be as above. Let M_{ij} and N_{ij} := p(M_{ij}, g_{ij}) be continuous symmetric tensors on M^{n} ×[0, T], where M_{ij} is of the class C^{2,1} for t > 0 and the zero eigenvector condition holds for N_{ij}. Furthermore, let the vector fields u^{k}, w^{k} be bounded on M^{n} × [0, T] and ∂M^{n} × [0, T] respectively with g(u, μ) ≥ 0, g(w, μ) > 0 on the boundary. Suppose the following conditions are fulfilled (δ > 0 being an arbitrary, small constant):

(i)  M_{ij}(., 0) ≥ 0,
(ii)  L(M_{ij}) ≥ 0 on M^{n} × (0, T],
(iii) (∇_{w} M_{ij})v^{i}v^{j}(x, t) ≥ 0 ∀(x, t) ∈ ∂M^{n}×(0, T], ∀ minimal eigen-
vectors v of M_{ij} with eigenvalue ∈ [−δ, 0].

Then M_{ij} ≥ 0 on M^{n} × [0, T].

Proof: We let M̂_{ij} := M_{ij} + ε e^{αt} g_{ij} , where 0 < ε < δ e^{−αT} and α > 0 will be chosen later, and show that M̂_{ij} > 0 on M^{n} × [0, T]. The Theorem then follows by ε → 0.

Suppose there exists a first zero eigenvector v of M̂_{ij} at a point (x_{0}, t_{0}) ∈ M^{n} × (0, T]. Extend v to a local vector field in a neighbourhood of (x_{0}, t_{0}) such that

∇v(x_{0}, t_{0}) = 0 , \frac{∂}{∂t}v(x_{0}, t_{0}) = 0 ,

and define a scalar function f by

f(x, t) := (M̂_{ij}v^{i}v^{j})(x, t).

Similarly as in [9], for sufficiently large values of α, f fulfils the inequality

(∗)  \frac{∂f}{∂t} (x_{0}, t_{0}) > (Δf + u^{k} ∇_{k} f)(x_{0}, t_{0}) ,

and f has a first zero at (x_{0}, t_{0}). Now x_{0} cannot be an interior point, for then (∗) yields \frac{∂f}{∂t} (x_{0}, t_{0}) > 0 in contradiction to the minimality of t_{0}. Hence x_{0} ∈ ∂M^{n}, and f(x, t_{0}) > 0 in the interior, i.e. f(., t_{0}) has a local minimum at x_{0}. Thus for all tangential directions τ ∈ T_{x_{0}}∂M^{n} we have 0 = ∇_{τ} f(x_{0}, t_{0}), and by assumption (iii) and g(w, μ) ≥ 0 we conclude

(∗∗)  ∇_{μ} M_{ij}(x_{0}, t_{0})v^{i}v^{j} ≥ 0.

Since g(u, μ) ≥ 0, this implies u^{k}∇_{k} M_{ij}(x_{0}, t_{0})v^{i}v^{j} ≥ 0 and from (∗) we get

\frac{∂f}{∂t} (x_{0}, t_{0}) > Δf(x_{0}, t_{0}) .

Now \frac{∂f}{∂t} (x_{0}, t_{0}) ≤ 0 by the minimality of t_{0}, and so we have Δf(., t_{0}) < 0 even in a small neighbourhood of x_{0} on M^{n}. Therefore ∇_{μ} f(x_{0}, t_{0}) < 0 by the elliptic Hopf Lemma and in contradiction to (∗∗). □

In applications, the following Lemma will be useful:

Lemma 3.4 Under the assumptions of Theorem 3.3 suppose that for (x, t) ∈ ∂M^{n} × [0, T] the tensor M_{ij} decomposes, i.e.

M_{Iμ}(x, t) = M_{μI}(x, t) = 0 ∀ 1 ≤ I ≤ (n−1).

Then condition (iii) of Theorem 3.3 can be replaced by

(iii′) (∇_{w} M_{ij})v^{i}v^{j}(x, t) ≥ 0 ∀ minimal eigenvectors v ∈ T_{x}∂M^{n}
of M_{ij} with eigenvalue ∈ [−δ, 0] ,
(iii)′′ (∇_{w} M_{μμ})(x, t) ≥ 0 if M_{μμ} min. and − δ ≤ M_{μμ}(x, t) ≤ 0 .

Proof: If at a boundary point v := v^{i}∂_{i} denotes a minimal eigenvector then, due to the block form of M_{ij}(x, t), the vectors v′ := v^{I}∂_{I} ∈ T_{x}∂M^{n} and v′′ := v^{n}μ are also minimal eigenvectors; at least one of these two vectors is nonzero, and so working with this vector instead of v and conditions (iii)′, (iii)′′ instead of (iii) in the proof of Theorem 3.3, we obtain the desired result. □

4 Barriers

In this section, we study the ”C^{0}-behaviour” of hypersurfaces moving under mean-curvature-like flows. To this end, let F : M^{n} × [0, T] ⟶ IR^{n+1} be a solution to equation 1.2 ; furthermore, let M_{B}^{n} be a compact, orientable manifold with boundary, and

F_{B} : M_{B}^{n} × [0, T] ⟶ IR^{n+1}, B_{t} := F_{B}(M_{B}^{n}, t)

a family of smooth imbeddings with ∂B_{t} ⊂ Σ ∀t such that B_{t} and Σ bound a domain G_{t} ⊂ IR^{n+1}. Assume that F_{B} obeys the following initial-boundary-value problem:

(4.1)  \{ \begin{matrix}⟨\frac{d}{dt}F_{B}, ν_{B}⟩(x, t) & ≥ & −H_{B}(x, t) ∀(x, t) ∈ M_{B}^{n} × (0, T] , \\ F_{B}(., 0) & = & F_{B,0} , \\ ∂B_{t} & ⊂ & Σ , \\ ⟨ν_{B}, μ⟩ & ≤ & 0 on ∂M_{B}^{n} × [0, T] . \end{matrix}

Here ν_{B} denotes the exterior unit normal field to B_{t} with respect to the domain G_{t}. Thus the equation means that the normal component of the speed which determines the motion of the B_{t} is not greater than it would be under the mean curvature flow. B_{t} is called a barrier for the M_{t}, if M_{0} ⊂ Ḡ_{0}\B_{0} implies M_{t} ⊂ Ḡ_{t}\B_{t} ∀t ∈ [0, T).

The main result of this section is

Theorem 4.1 Let F_{B} be a family of smooth imbeddings as above, and let M_{0} ⊂ Ḡ_{0}\B_{0}. Then B_{t} is a barrier for M_{t}.

Before proving and discussing this Theorem in detail, we have to do some preparatory work. We thereby leave the tedious, but nevertheless elementary calculations to the reader.

In the sequel, let t_{0} > 0 be the first time at which there exists p_{0} ∈ M_{t_{0}} ∩B_{t_{0}}. Then clearly

T_{p_{0}}M_{t_{0}} = T_{p_{0}}B_{t_{0}} ,

for a transverse intersection of M_{t_{0}} and B_{t_{0}} would contradict the minimality of t_{0}. (By the boundary conditions for M_{t} and B_{t}, this holds equally if p_{0} ∈ ∂M_{t_{0}}.)

We introduce generalized Gaussian coordinates in a tubular neighbourhood U_{ε} of M_{t_{0}} as in section 2. If we thereby choose ξ and ν such that ν(p_{0}) = ξ(p_{0}) = ν_{B}(p_{0}), we have x^{n+1}(p) < 0 if and only if p ∈ G. Now, cutting off the hypersurfaces B_{t} at the boundary of U_{ε} and choosing δ > 0 sufficiently small, we obtain graph representations u, w : M^{n} × [t_{−}, t_{+}] ⟶ IR of the cutted B_{t} and of M_{t} on the time interval [t_{−}, t_{+}] := [t_{0} − δ, t_{0} + δ] ∩ [0, T]. To be more precise, we let u(x, t) be the x^{n+1}-coordinate of the intersection point of the flux line through F (x, t_{0}) on B_{t}, if such an intersection point exists in U_{ε}; otherwise, we set u(x, t) := ±ε, choosing the sign such that u is a Lipschitz function.

Next, we define φ := u − w and observe that φ > 0 on [t_{−}, t_{0}). Moreover, choosing δ > 0 such that M_{t} ⊂ U_{ε/10} on [t_{−}, t_{+}], the above construction implies that φ is smooth in a neighbourhood of any (x_{1}, t_{1}) ∈ M^{n} × [t_{−}, t_{+}] with |φ(x_{1}, t_{1})| < ε/2.

As in section 2, the geometric boundary condition ⟨ν_{B}, μ⟩ ≤ 0 implies the scalar boundary condition μ^{k}D_{k}u ≥ μ^{n+1} (in the neighbourhood of a boundary point (x_{1}, t_{1}) with |φ(x_{1}, t_{1})| < ε/2), where μ^{α} = μ^{α}(x, t, u) are the components of μ with respect to the chosen coodinates. Analogously, μ^{k}D_{k}w = μ^{n+1} with μ^{α} = μ^{α}(x, t, w), and hence

0 ≤ μ^{k}(x, t, u) D_{k}u − μ^{k}(x, t, w) D_{k}w − μ^{n+1}(x, t, u) + μ^{n+1}(x, t, w)
= ∫_{0}^{1} \frac{d}{ds} (μ^{k}(x, t, su + (1−s)w) · (sD_{k}u + (1−s)D_{k}w)) ds
− ∫_{0}^{1} \frac{d}{ds} (μ^{n+1}(x, t, su + (1−s)w)) ds
= [∫_{0}^{1} μ^{k}(x, t, .) ds] · D_{k}φ
+ [∫_{0}^{1} (D_{z}μ^{k}(x, t, .) · (sD_{k}u + (1−s)D_{k}w) − D_{z}μ^{n+1}(x, t, .)) ds] · φ
=: v^{k}(x, t) · D_{k}φ + Ψ(x, t) · φ .

Hereby the functions v^{k} and f(φ, x, t) := −Ψ(x, t) · φ are smooth, and, by the mean value Theorem,

v^{k}(x, t) = ∫_{0}^{1} μ^{k}(x, t, su + (1−s)w) ds = μ^{k}(x, t, σ) , |σ| < ε

near (x_{1}, t_{1}). By the compactness of ∂M^{n}, v^{k} will therefore be a tranversal vector field for sufficiently small ε > 0. Thus we have proved:

Lemma 4.2 Let (x_{1}, t_{1}) ∈ ∂M^{n} × [t_{−}, t_{+}] with |φ(x_{1}, t_{1})| < ε/4. Then in a neighbourhood of (x_{1}, t_{1}), the boundary condition

D_{v}φ(x, t) = f(φ, x, t)

is fulfilled for φ. Hereby, f is a smooth function with

|f(s, x, t) − f(0, x, t| ≤ const · |s| , f(0, x, t) ≥ 0 ,

and v ∈ T M^{n} is a smooth, outward pointing vector field, i.e. g(v, μ) > 0. □

4.1 Proof of the Theorem and Further Results

Proof of Theorem 4.1: Let t_{0} ∈ (0, T] be as above; we have to show that t_{0} = T. Suppose t_{0} < T, and choose δ > 0, ε > 0 such that u, w are graph representations of B_{t}, M_{t} as before. Then, if φ = u − w, the choice of t_{0} implies φ(., t) > 0 on [t_{−}, t_{0}), and φ(x_{0}, t_{0}) = 0 for some x_{0} ∈ M^{n}. As in section 2, we have parabolic differential equations for u and w:

\begin{aligned}\frac{∂u}{∂t} & ≥ & −v_{B} · H_{B} & = & α^{ij}(x, t, u, Du)D_{i}D_{j}u + γ(x, t, u, Du) , \\ \frac{∂w}{∂t} & = & −v · H & = & α^{ij}(x, t, w, Dw)D_{i}D_{j}w + γ(x, t, w, Dw) . \end{aligned}

By standard methods, this implies a linear parabolic differential equation for φ = u − w:

\frac{∂φ}{∂t} ≥ a^{ij}D_{i}D_{j}φ + b^{i}D_{i}φ + cφ ,

with a^{ij} = a^{ij}(x, t), b^{i} = b^{i}(x, t), c = c(x, t) . Using Lemma 4.2, we see that φ fulfils the conditions of Theorem 3.1 (weak maximum principle); hence φ(., t) ≥ 0 on [t_{−}, t_{+}]. By the strong maximum principle (Theorem 3.2), φ(., t) > 0 on [t_{−}, t_{+}) in contradiction to φ(x_{0}, t_{0}) = 0. □

Remark 4.3 If M_{0} ∩ B_{0} ≠ ∅ and M_{0} ≢ B_{0}, the strong maximum principle yields that M_{t} and B_{t} separate immediately, i.e. M_{t} ∩ B_{t} = ∅ ∀t > 0.

Theorem 4.1 implies the case of ”static” barriers: Let \frac{d}{dt}F_{B} ≡ 0 and B := F_{B,0}(M_{B}^{n}). Furthermore, let H_{B} ≥ 0. Then F_{B} fulfils equation 4.1, and Theorem 4.1 yields

Corollary 4.4 (”Static” Barriers) If H_{B} ≥ 0 and ⟨ν_{B}, μ⟩ ≤ 0 on the boundary, B is a barrier for M_{t}. □

Here we have not made any assumptions on M_{0}; instead, we have required H_{B} ≥ 0 for the mean curvature of the barrier. This condition can be weakened if we impose a condition on the curvatures of M_{0} and Σ: If H(., 0) ≥ C > 0 and Σ is assumed to be convex with respect to the outward unit normal field, it is sufficient to require H_{B} ≥ −C for the mean curvature of B. The proof is very similar to the above techniques; convexity of Σ hereby is needed to guarantee that the relation H ≥ C is preserved for t > 0 (cf. Theorem 8.1).

We remark that the above results can be generalized in a straightforward manner to domains which are bounded by Σ and several barriers.

Corollary 4.5 (Imbeddings) Let ^{Σ}H ≥ 0 with respect to the outward unit normal field μ. Then M_{0} being an imbedding implies that all M_{t} are imbeddings as well.

Proof: We must show that M_{t} cannot selfintersect. Suppose t_{0} is the first time where a selfintersection occurs at points p = F (x_{0}, t_{0}), q = F (y_{0}, t_{0}). By Theorem 4.1 we can exclude the cases x_{0}, y_{0} ∈ M^{n}\∂M^{n} and x_{0}, y_{0} ∈ ∂M^{n}; thus it remains to rule out the case x_{0} ∈ M^{n}\∂M^{n}, y_{0} ∈ ∂M^{n}. But as ^{Σ}H ≥ 0, the hypersurface Σ is a barrier to M_{t} (according to Corollary 4.4), and hence

x ∉ ∂M^{n} ⇔ F (x, 0) ∉ Σ ⇔ F (x, t) ∉ Σ ∀t < T .

Thus F (x_{0}, t_{0}) ≠ F (y_{0}, t_{0}) by the choice of x_{0} and y_{0}. □

We remark that the condition ^{Σ}H ≥ 0 is required only to prevent M_{t} from intersecting Σ.

5 Distance of Hypersurfaces

In section 4 we considered families B_{t} of hypersurfaces such that a solution of equation 1.2 could never intersect the B_{t}. Now we are interested in discussing the behaviour of the distance between two solutions. To this end, let M_{1}^{n}, M_{2}^{n} be compact, smooth, orientable manifolds. The main Theorem of this section is

Theorem 5.1 Let G be a convex domain in IR^{n+1} with smooth boundary Σ = ∂G. Let F_{i} : M_{i}^{n} × [0, T] ⟶ IR^{n+1} be two families of smooth immersions moving in Ḡ under the mean curvature flow equation 1.2; hereby, defining M_{i}(t) := F_{i}(M_{i}^{n}, t), let M_{1}(0) ∩ M_{2}(0) = ∅. Then the function d(t) := dist_{n+1}(M_{1}(t), M_{2}(t)) is monotonically increasing.

As in the last section, the main idea of the proof is to consider graph representations of M_{i}(t) in the neighbourhood of ”critical points”, i.e. points p_{i} ∈ M_{i}(t) with d(t) = ∥p_{1} −p_{2}∥ and applying the maximum principles of section 3. Again, we first have to do some preparatory work: Throughout this section, let G ⊂ IR^{n+1} be a convex domain with smooth boundary Σ and exterior unit normal field μ. Given p, q ∈ Ḡ, p ≠ q, define

ω_{pq} := \frac{p−q}{∥p−q∥} .

The following Lemma is a simple consequence of the convexity of G:

Lemma 5.2 (Convexity of G)

(i) Let p ∈ Σ , q ∈ Ḡ , p ≠ q. Then ⟨μ(p), ω_{pq}⟩ ≥ 0.

(ii) Let p, q ∈ Σ , p ≠ q , ⟨μ(p), ω_{pq}⟩ = ⟨μ(q), ω_{pq}⟩ = 0. Then ⟨μ(p), μ(q)⟩ = 1, and p, q belong to the same hyperplane. □

We now consider smooth immersions F_{i} of M_{i}^{n} (i = 1, 2) in IR^{n+1} with unit normal fields ν_{i}. Let M_{i} := F_{i}(M_{i}^{n}) ⊂ Ḡ, M_{1} ∩ M_{2} = ∅, ∂M_{i} ⊂ Σ and ⟨ν_{i}, μ⟩ = 0 on ∂M_{i}. Moreover, let p := F_{1}(x_{1}) and q := F_{2}(x̃_{2}) such that d := ∥p − q∥ = dist_{n+1}(M_{1}, M_{2}) . We then introduce Cartesian coordinates {y^{1}, ..., y^{n+1}} in IR^{n+1} such that y(p) = (0, ..., 0, d/2), y(q) = (0, ..., 0, −d/2). In a neighbourhood of p, q, the hypersurfaces M_{i} can be written as graphs of smooth functions u_{i} with u_{1}(0) = d/2, u_{2}(0) = −d/2 and Du_{1}(0) = Du_{2}(0) = 0. This can be seen as follows: By the choice of p, q, the function

f : M_{1}^{n} ⟶ IR, f(x) := ∥F_{1}(x) − q∥^{2}

has a global minimum in x_{1}. Now, if p ∉ Σ, we have ∇f(x_{1}) = 0 and thereby Du_{1}(0) = 0. If p ∈ Σ, we have ⟨∇f(x_{1}), w⟩ = 0 for all tangential directions w ∈ T ∂M_{1}(p), and

0 ≥ ⟨∇f(x_{1}), μ⟩ = 2 ⟨F_{1}(x_{1}) − q, μ⟩ = 2 ⟨p − q, μ⟩ .

Using Lemma 5.2(i), this implies ⟨∇f(x_{1}), μ⟩ = 0 and thus Du_{1}(0) = 0 as before.

Lemma 5.3 The matrix (D_{i}D_{j}(u_{1} − u_{2}))(0) is positive semidefinite.

Proof: Let u_{i} : Ω_{i} ⟶ IR with 0 ∈ \overline{Ω_{i}}, and let φ := u_{1} − u_{2} on Ω_{1} ∩ Ω_{2}. If p, q ∉ Σ, 0 is in the interior of Ω_{i}, and the result follows by Taylor expansion of φ. The same argument applies if p ∉ Σ and q ∈ Σ (or vice versa), or if p, q ∈ Σ and T_{0}∂Ω_{1} = T_{0}∂Ω_{2}. Finally the case p, q ∈ Σ and T_{0}∂Ω_{1} ≠ T_{0}∂Ω_{2} can be ruled out as, according to Lemma 5.2(ii), we have ⟨μ(p), μ(q)⟩ = 1 if p, q ∈ Σ. □

Finally, we state a Lemma due to Hamilton [10]:

Lemma 5.4 Let g : M^{n} × [0, T] ⟶ IR be a smooth function, and define f(t) := min_{M^{n}} g(x, t). Then f is a Lipschitz function, and

\frac{df}{dt} (t) ≥ min\{\frac{∂g}{∂t} (x̂, t) : x̂ ∈ M^{n} such that g(x̂, t) = min_{x∈M^{n}} g(x, t)\} .

5.1 Proof of the Theorem and Further Results

Proof of Theorem 5.1: Let

g : M_{1}^{n} × M_{2}^{n} × [0, T] ⟶ IR , g(x, x̃, t) := ∥F_{1}(x, t) − F_{2}(x̃, t)∥^{2} ,

then d(t) = min_{M_{1}^{n}×M_{2}^{n}} g(x, x̃, t), and, using the evolution equation 1.2, we have

\frac{d}{dt}g(x, x̃, t) = 2 ⟨F_{1}(x, t)−F_{2}(x̃, t), H_{1}(x, t)−H_{2}(x̃, t)⟩ .

Now let t_{0} ∈ [0, T), p = F_{1}(x_{1}, t_{0}), q = F_{2}(x̃_{2}, t_{0}) be such that d(t_{0}) = ∥p − q∥ > 0 and use the Cartesian coordinates defined above to conclude

H_{1}(x_{1}, t_{0}) − H_{2}(x̃_{2}, t_{0}) = Δ(u_{1} − u_{2})(0, t_{0}) · ω_{pq} ,
F_{1}(x_{1}, t_{0}) − F_{2}(x̃_{2}, t_{0}) = d(t_{0}) · ω_{pq} .

Lemma 5.3 yields Δ(u_{1} − u_{2})(0, t_{0}) ≥ 0, and thus \frac{d}{dt}g(x_{1}, x̃_{2}, t_{0}) ≥ 0. The result follows from Lemma 5.4. □

Let us, for a moment, return to the study of barriers: G and Σ being defined as above, we let B := F_{B}(M_{B}^{n}) ⊂ Ḡ be a smooth imbedding with unit normal field ν_{B} , ∂B ⊂ Σ and ⟨ν_{B}, μ⟩ = 0 on ∂B. We assume that G is divided by B into two disjoint subdomains G_{1}, G_{2}, and that ν_{B} is the exterior normal field to G_{1}. In this situation, we have

Theorem 5.5 Let F be a family of smooth immersions evolving under the mean curvature flow equation 1.2, and M_{0} ⊂ Ḡ_{1}. Let H_{B} ≥ 0 with respect to ν_{B}. Then the function d(t) := dist_{n+1}(M_{t}, B) is monotonically increasing.

Proof: Defining g(x, x̃, t) := ∥F_{B}(x)−F (x̃, t)∥^{2} and p, q, x_{1}, x̃_{2} etc. as before, we have

\frac{d}{dt}g(x_{1}, x̃_{2}, t_{0}) = 2 d(t_{0}) ⟨ω_{pq}, 0 − H(x̃_{2}, t_{0})⟩ .

Using H_{B} ≥ 0 and ω_{pq} = ν_{B}(x_{1}),

\frac{d}{dt}g(x_{1}, x̃_{2}, t_{0}) ≥ 2 d(t_{0}) ⟨ω_{pq}, H_{B}(x_{1}, t_{0}) − H(x̃_{2}, t_{0})⟩,

and, proceeding as in the proof of Theorem 5.1, we obtain the result. □

There is a variety of configurations which can be studied using essentially the same techniques as presented here; for example, it is possible to consider imbedded hypersurfaces moving under

⟨\frac{d}{dt}F , ν⟩(x, t) ≥ −H(x, t), ⟨ν, μ⟩ = 0 on ∂M^{n} .

Here we must be careful with defining exterior normal fields, for the evolution inequality is not invariant under the transformation ν → −ν. However, if we assume G to be a domain bounded by Σ and M_{1}(0), M_{2}(0) and define ν_{i} as exterior normal fields to G, we can conclude that the distance between M_{1}(t) and M_{2}(t) once more monotonically increases. A proof of this fact is given in [23].

6 A Local Gradient Bound

In section 2, we saw that locally equation 1.2 is equivalent to the scalar problem 2.1. Now, by a precise control of the gradient Dw, the latter equation will be (locally) uniformly parabolic; from there, by standard results of the linear theory the existence Theorem 1.3 follows. In order to establish the local gradient bound, we assume that the curvature of the M_{t} already is under control on an arbitrary closed time interval [0, T] ⊂ [0, T_{C}). We caution the reader to remark that the crucial point is to obtain a local gradient estimate which is uniform on some sufficiently large time interval; this is certainly not a straightforward consequence of the underlying curvature bound.

Assumption 6.1 Throughout this section, let F : M^{n} × [0, T] ⟶ IR^{n+1} be a solution to equation 1.2, and let the curvature of M_{t} be uniformly bounded on [0, T] as follows:

|g^{ij} h_{jk}|(x, t) ≤ (2 n^{2}ρ_{0})^{−1} ∀1≤i, k ≤n , ∀(x, t) ∈ M^{n} × [0, T] ,

where ρ_{0} is a positive constant. Furthermore, let Σ be smooth with a global curvature bound, i.e. there exists ρ_{Σ} > 0 such that

|^{Σ}g^{ij} h_{jk}^{Σ}|(p) ≤ (2 n^{2}ρ_{Σ})^{−1} ∀1≤i, k ≤n , ∀p ∈ Σ .

Finally, let Σ ∩ B_{ρ_{Σ}}^{n+1}(p) be connected for all p ∈ Σ.

We emphasize that in order to obtain the local gradient bound on a small region in spacetime, it is sufficient to require a local curvature bound to the M_{t} on a (slightly larger) region; however, the global curvature bound somewhat simplifies the notation. The global curvature bound on Σ is a very slight restriction, as was discussed in Remark 1.2. Finally, the connectedness of Σ ∩B_{ρ_{Σ}}^{n+1}(p) is included for technical reasons; in general, this requirement can be fulfilled by choosing ρ_{Σ} sufficiently small, i.e. working in sufficiently small domains.

Now let us fix some notation: By the standard graph representation (SGR) w of M_{t} near a given point p_{0} := F (x_{0}, t_{0}) we mean the local graph representation around p_{0} with respect to a Cartesian coordinate system in IR^{n+1} such that p_{0} = (0, w(0, t_{0})) = (0, 0) and ν(x_{0}, t_{0}) =: ω = e_{n+1} (i.e. Dw(0, t_{0}) = 0). Furthermore, given ω ∈ IR^{n+1}, we define

u : M^{n} × [0, T] ⟶ IR , u(x, t) := ⟨F (x, t), ω⟩;

thus u(x, t) describes the height of a point F (x, .) over the hyperplane H through F (x_{0}, t_{0}), whereas w(y, .) is the height of a point with fixed coordinates on H.

Z_{r}(p) := B_{r}^{n}(p) × (−∞, +∞) ⊂ IR^{n+1} denotes the space cylinder around p with axis ω; analogously Z_{r,C}(p) := B_{r}^{n}(p) × (−C r, +C r).

Σ_{t_{0}}^{Δ} := {F (x, t) : x ∈ ∂M^{n}, |t − t_{0}| ≤ Δ} is the subset of Σ which coincides with the boundaries ∂M_{t} in the time interval [t_{0} − Δ, t_{0} + Δ].

6.1 Hypersurfaces with Bounded Curvature

In order to get the right scaling behaviour of the desired estimates, we first have to state some facts concerning hypersurfaces M := F (M^{n}) ⊂ IR^{n+1} with a curvature bound

|g^{ij} h_{jk}|(x) ≤ (2 n^{2}ρ)^{−1} ∀1≤i, k ≤n , ∀x ∈ M^{n}

as in Assumption 6.1.

Lemma 6.2 Let p_{0} = F (x_{0}, t_{0}) and w : IR^{n} ∋ U ⟶ IR be the SGR of F (M^{n}) around p_{0}. Let p = F (x) = (y, w(y)) with ∥y∥ =: r < ρ and {s y : 0 ≤ s ≤ 1\} ⊂ U. Finally, define v(x) = ⟨ν(x), ω⟩^{−1} = \sqrt{1 + ∥Dw∥^{2}}(y). Then

v(x) < 1 + (\frac{r}{2 ρ})^{2} , ⟨ν(x), ω⟩ > 1 − (\frac{r}{2 ρ})^{2} , |⟨F (x) − F (x_{0}), ω⟩| < \frac{r^{2}}{ρ} .

Proof (sketch): From the curvature bound we get |h_{ik}| ≤ \frac{1}{2 n ρ} (1 + ∥Dw∥^{2}) and thereby

|D_{i}D_{k}w| ≤ (2 nρ)^{−1} \sqrt{1 + ∥Dw∥^{2}}^{3} .

Now let e := y/∥y∥; then, by assumtion, the function σ ⟼ D_{i}w(σ e) is defined on [0, 1]. From Hölder’s inequality and the above equation, we obtain

|D_{i}w(s e)| ≤ \sqrt{\frac{r}{n}} \frac{1}{2 ρ} \sqrt{∫_{0}^{s}(1 + ∥Dw∥^{2})^{3}dσ}

and by summation

(1 + ∥Dw∥^{2})(s e) ≤ 1 + \frac{r}{4 ρ^{2}} ∫_{0}^{s}(1 + ∥Dw∥^{2})^{3}(σ e) dσ .

Solving this ordinary integral equation for 1 + ∥Dw∥^{2} and using Taylor expansions, the result follows. □

Note that the Lemma implies vice versa, that the SGR of M will be defined on the entire ball B_{r}^{n}(0), if the p_{0}-component of Z_{r}(p_{0}) ∩ M does not have any boundary points in the interior of Z_{r}(p_{0}). Thus, applying the Lemma to Σ and recalling Remark 1.2, we arrive at the following

Lemma 6.3 Let r ≤ ρ_{Σ}, p_{0} = F (x_{0}, t_{0}) ∈ Σ, γ := μ(p_{0}) and v_{Σ} := ⟨μ, γ⟩^{−1}. Then for any p ∈ Σ ∩ Z_{r}(p_{0}), we have

v_{Σ}(p) < 1 + (\frac{r}{2 ρ_{Σ}} )^{2} , ⟨μ(p), γ⟩ > 1 − (\frac{r}{2 ρ_{Σ}} )^{2} , |⟨p − p_{0}, γ⟩| < \frac{r^{2}}{ρ_{Σ}} . □

In order to obtain an analogous estimate for the hypersurfaces M_{t}, we have to be more careful: If p_{0} ∈ M_{t_{0}}\∂M_{t_{0}} we can proceed exactly as before, obtaining a graph representation on an entire ball of radius ρ_{1} := min{ρ/4, d/2}, where d := dist_{n+1}(p_{0}, Σ). If p_{0} ∈ ∂M_{t_{0}}, due to the curvature of Σ, the SGR of M_{t_{0}} will in general not be defined along straight lines σ ↦ σ e as in the proof of Lemma 6.2. However, the curvature of Σ being bounded, we have a graph representation for Σ if r ≤ ρ_{Σ}, and it turns out that we can work with curves of length ≤ 3 r to connect various points in the domain of the SGR of M_{t_{0}}. Thus we obtain

Lemma 6.4 Let p_{0} = F (x_{0}, t_{0}) , d := dist_{n+1}(p_{0}, ∂M_{t_{0}}),

ρ_{1} := \{ \begin{matrix}min\{ρ_{0}/3 , ρ_{Σ} \} & if p_{0} ∈ Σ , \\ min\{ρ_{0}/4 , d/2\} & if p_{0} ∉ Σ , \end{matrix}

and r ≤ ρ_{1}. Then for any F (x, t_{0}) ∈ Z_{r}(p_{0}) we have:

v(x, t_{0}) < 1+(\frac{r}{ρ_{0}} )^{2} , ⟨ν(x, t_{0}), ω⟩ > 1−(\frac{r}{ρ_{0}} )^{2} , |u(x, t_{0})−u(x_{0}, t_{0})| < \frac{2 r^{2}}{ρ_{0}} .

□

Certainly we could have chosen ρ_{1} := δ·min{ρ_{0}, ρ_{Σ}} (δ > 0 being sufficiently small) for both interior points and points near the boundary, but as we want to derive the inner and boundary gradient estimates independently, the above definition is somewhat simpler from the technical point of view.

As a further consequence of the evolution equation 1.2 and the curvature bound, we can control the behaviour of points F (x_{0}, t) for fixed x_{0} ∈ M^{n}: A short calculation yields for any t_{1}, t_{2} ∈ [0, T]:

(∥F ∥^{2} − u^{2})(x, t_{2}) ≤ (\frac{1}{2 ρ_{0}} |t_{2} − t_{1}| + \sqrt{∥F ∥^{2} − u^{2}} (x, t_{1}))^{2} . (6.1)

Combining this equation with the above Lemmata, we can estimate the distance of points F (x, t) , F (x_{0}, t_{0}), if F (x, t_{0}) ⊂ Z_{r}(p_{0}). These points will leave the cylinder Z_{r}(p_{0}) after some time; on the other hand, Z_{r}(p_{0}) contains points F (x, t) which didn’t start in the cylinder for t = t_{0}. Now we are interested in controlling the behaviour of M_{t} ∩ Z_{r}(p_{0}) for a short time |t − t_{0}| ≤ δ. Combining the above results, we easily see that this can be done on a smaller cylinder Z_{r_{2}}(p_{0}) ⊂ Z_{r}(p_{0}) as follows:

Lemma 6.5 Let r_{2} ≤ \frac{ρ_{1}}{2} and δ ≤ 2 ρ_{0} r_{2} . Then for any (x, t) ∈ M^{n} × [0, T] with F (x, t) ∈ Z_{r_{2}}(p_{0}) and |t − t_{0}| ≤ δ the following estimate holds:

|u(x, t) − u(x_{0}, t_{0})| < \frac{1}{2 ρ_{0}} (|t − t_{0}| + 16 r_{2}^{2}) . □

As a last consequence of the curvature bound on the M_{t} we give an estimate for the distance of a point p from Σ_{t_{0}}^{Δ}. The proof follows directly from equation 1.2.

(6.2) dist_{n+1}(p, Σ_{t_{0}}^{Δ}) ≥ dist_{n+1}(p, ∂M_{t_{0}}) − \frac{1}{2 ρ_{0}} Δ .

6.2 The Interior Estimate

We first consider p_{0} = F (x_{0}, t_{0}) with x_{0} ∈ M^{n}\∂M^{n}. As before, we thereby choose Cartesian coordinates in IR^{n+1} with origin at p_{0} and ω := e_{n+1} := ν(x_{0}, t_{0}). Using ideas of Ecker and Huisken [5], we define

M_{r,δ} := sup{|u(x, t)| : F (x, t) ∈ Z_{r}(p_{0}) , 0 ≤ t − t_{0} ≤ δ} ,

φ(x, t) := \{ \begin{matrix}(\frac{r}{2 β} (u−M_{r,δ}) + r^{2} − (∥F ∥^{2}−u^{2}))_{+} (x, t) , & F (x, t) ∈ Z_{r}(p_{0}) , \\ 0 , & F (x, t) ∉ Z_{r}(p_{0}) , \end{matrix}

η(x, t) := e^{λ φ(x,t)} − 1 ,

where r > 0 , δ > 0 , β > 0 and λ > 0 will be chosen later. φ and η are Lipschitz functions for 0 ≤ t − t_{0} ≤ δ, and supp(φ) = supp(η) ⊂ Z_{r}(p_{0}). For the sequel, we fix the following notation:

sup_{Z_{r}} v(., t_{0}) := sup{v(ξ, t_{0}) : F (ξ, t_{0}) ∈ Z_{r}(p_{0})} ,
sup_{Z_{r}, δ} η(., .) := sup{η(ξ, τ) : F (ξ, τ) ∈ Z_{r}(p_{0}), 0 ≤ t − t_{0} ≤ δ} .

Now let (x_{1}, t_{1}) , x_{1} ∈ M^{n}\∂M^{n} , 0 ≤ t_{1}−t_{0} ≤ δ be a point where η v attains a new maximum. Proceeding as in [5], we obtain

2 n ≥ λ r^{2} (\frac{1}{4 β^{2}} (1 − v^{−2}) − \frac{2}{β} v^{−1})

at this point. Hence, defining λ := \frac{8 n α β^{2}}{r^{2}} (α > 1 to be chosen), we conclude

v(x_{1}, t_{1}) < \frac{α}{α−1} (8 β + 1) .

This in turn yields

(η v)(x_{1}, t_{1}) < sup_{Z_{r}, δ} η(., .) · max \{ \frac{α}{α−1} (8 β + 1) , (1 + ε̂) sup_{Z_{r}} v(., t_{0})\} ,

where ε̂ > 0 is an arbitrary constant. Now this inequality can only be broken at a boundary point x_{2} ∈ ∂M^{n}; hence an interior bound for η v can be obtained by choosing r > 0 and δ > 0 such that the p_{0}-components of M_{t} ∩ Z_{r}(p_{0}) do not have any boundary points for 0 ≤ t − t_{0} ≤ δ. Using Lemma 6.5 and equation 6.2, by a short calculation we see that r ≤ min{ρ_{0}/8, d/8} and δ ≤ 2 ρ_{0} r are sufficiently small (d = dist_{n+1}(p_{0}, ∂M_{t_{0}})). Hence, from

sup_{Z_{r}, δ} η(., .) ≤ e^{8 n α β^{2}} − 1 , (6.3)

we conclude

Lemma 6.6 Let p_{0} = F (x_{0}, t_{0}) , x_{0} ∈ M^{n}\∂M^{n} , d := dist_{n+1}(p_{0}, ∂M_{t_{0}}) > 0 and α > 1 , β > 0 , ε̂ > 0 , r ≤ min\{\frac{ρ_{0}}{8} , \frac{d}{8}\} , δ ≤ 2 ρ_{0} r . Then for any (x, t) with F (x, t) ∈ Z_{r}(p_{0}) and 0 ≤ t − t_{0} ≤ δ, the following estimate holds:

(η v)(x, t) < (e^{8 n α β^{2}} − 1) · max \{ \frac{α}{α−1} (8 β + 1) , (1 + ε̂) sup_{Z_{r}} v(., t_{0})\} . □

Lemma 6.6 immediately yields an estimate for v by bounding η from below. Certainly, this has to be done on a smaller cylinder Z_{r_{1}}(p_{0}), since η ≡ 0 on ∂Z_{r}(p_{0}). The crucial point is to determine the scaling behaviour of β, and it turns out that a good choice is

β := \sqrt[\frac{1}{2}]{\frac{r}{ρ_{0}}} .

If we then let κ ∈ (0, 1) , r ≤ (\frac{κ}{32})^{2} ρ_{0} , r_{1} ≤ \sqrt{\frac{κ}{4}} r and δ ≤ 8 r^{2}, we estimate

r^{2} ≥ φ(x, t) ≥ (1 − κ)r^{2}

in the cylinder Z_{r_{1}}(p_{0}) and for 0 ≤ t−t_{0} ≤ δ . This in turn implies the desired gradient estimate in the interior case on time interval which is comparable to ρ_{0}^{2} , d^{2} :

Theorem 6.7 Let p_{0} = F (x_{0}, t_{0}) , d = dist_{n+1}(p_{0}, ∂M_{t_{0}}) > 0. Then

∀ ε > 0 ∃ C_{1} > 0 , C_{2} > 0 , C_{3} > 0 , C_{4} > 0 depending only on n, ε

such that

v(x, t) < 1 + ε ∀(x, t) with F (x, t) ∈ Z_{r_{1}}(p_{0}) , 0 ≤ t − t_{0} ≤ δ;

hereby r_{1} = min{C_{1} ρ_{0} , C_{2} d} and δ = min{C_{3} ρ_{0}^{2} , C_{4} d^{2}}.

Proof: Choose

r := \frac{1}{n} min \{\sqrt{\frac{κ}{4}}d, (\frac{κ}{32})^{2}ρ_{0}\} , r_{1} := \sqrt{\frac{κ}{4}}r , δ ≤ min \{8 r^{2}, 2 ρ_{0} r\} .

Furthermore, let α := 1 + \frac{1}{κ} and ε̂:= κ. Then

(1 + ε̂) sup_{Z_{r}} v(., t_{0}) ≤ (1 + κ)^{2} (Lemma 6.4) and \frac{α}{α−1} (1 + 8 β) ≤ (1 + κ)^{2} .

Moreover, for (x, t) as in the Theorem, η(x, t) > e^{8 n α β^{2} (1−κ)} − 1. Now we apply the Taylor expansion

\frac{e^{z}−1}{e^{z (1−κ)}−1} ≤ 1 + 2 κ + 2 z , z ∈ (0, \frac{1}{32}) , κ ∈ (0, \frac{1}{2}]

to the estimate Lemma 6.6 and conclude

v(x, t) < 1 + 10 κ .

The result follows if we choose κ ≤ min\{\frac{1}{2}, \frac{ε}{10}\}. □

Remark 6.8 At a first glance, the estimate Theorem 6.7 depends on the point (x_{0}, t_{0}), since r, r_{1}, δ depend on the distance of F (x_{0}, t_{0}) from Σ. However, for all (x̃, t̃) with dist_{n+1}(F (x̃, t̃), Σ) ≥ d_{0} > 0, we obtain the same estimate, since r, r_{1}, δ depend only on the minimal distance of all F (x̃, t̃) from Σ.

Remark 6.9 It is clear that Theorem 6.7 immediately implies a local gradient estimate for the SGR of M_{t} in the vicinity of (x_{0}, t_{0}): Given ε > 0, there exist r̂_{1} > 0 and δ̂ > 0 such that the SGR of the M_{t} is defined on B_{r_{1}}(0) × [t_{0}, t_{0} + δ̂] and fulfils the estimate

\sqrt{1 + ∥Dw∥^{2}(y, t)} < 1 + ε ;

hereby r̂_{1}, δ̂ depend only on ε, ρ_{0}, d and have the ”natural” scaling behaviour r̂_{1} ∼ ρ_{0}, δ̂ ∼ ρ_{0}^{2}. Thus using Lemma 6.5, we obtain a local C^{1}-estimate

∥w∥_{C^{1}} ≤ C_{0} on Bn ; r̂_{1} × [t_{0}, t_{0} + δ̂].

6.3 The Boundary Estimate

We now turn our attention to boundary points. Here we have to rule out boundary maxima at least for sufficiently large values of η v. This can be done if we guarantee that ⟨∇(η v), μ⟩ < 0 for large values of η v. Now the cutoff-function η which we used in the interior case essentially is a paraboloid centered at p_{0}; therefore it does not produce a sufficiently large term ⟨∇η, μ⟩ < 0 that could compensate the (possibly large and positive) term ⟨∇v, μ⟩ in the vicinity of p_{0}. The idea is to center the paraboloid not at p_{0} but at a nearby point p_{m} ∉ Σ such that η posesses a sufficiently large gradient near p_{0}.

We choose Cartesian coordinates in IR^{n+1} with origin at p_{0} = F (x_{0}, t_{0}) and set ω := ν(x_{0}, t_{0}) , γ := μ(p_{0}) and

p_{m} := p_{0} − \frac{1}{K} r γ , K > 1 , r ≤ \frac{ρ_{Σ}}{4} .

Thus Z_{r,1}(p_{m}) ⊂ B_{ρ_{Σ}}^{n+1}(p_{0}), and hence Lemma 6.3 holds on Σ ∩ Z_{r,1}(p_{m}), and this set is connected due to Assumption 6.1. Now, similar to the interior case, we set (with Z_{r} := Z_{r}(p_{m}))

M_{r,δ} := sup{|u(x, t)| : F (x, t) ∈ Z_{r} , 0 ≤ t − t_{0} ≤ δ} ,

φ(x, t) := \{ \begin{matrix}(\frac{r}{2 β} (u−M_{r,δ}) + r^{2} − (∥F −p_{m}∥^{2}−u^{2}))_{+}(x, t) , & F (x, t)∈Z_{r}, \\ 0 , & F (x, t)∉Z_{r}, \end{matrix}

η(x, t) := e^{λ φ(x,t)} − 1 ,

where 0 < r < \frac{ρ_{Σ}}{4} , δ > 0 and α > 1 will be chosen later and

β := \sqrt[\frac{1}{2}]{\frac{r}{ρ_{0}}} , λ := \frac{8 n α β^{2}}{r^{2}} = \frac{2 n α}{ρ_{0} r} .

Now we estimate the normal component of ∇(η v) as follows:

Lemma 6.10 Let K ≥ 8 , r ≤ \frac{1}{64} min\{ρ_{0} , \frac{ρ_{0}}{K^{2}} , \frac{ρ_{0}}{n α} , \frac{ρ_{Σ}}{K} , \frac{ρ_{Σ}^{2}}{K^{2} ρ_{0}} \} and δ ≤ r^{2}. Then for any (x, t) ∈ M^{n} × [0, T] with F (x, t) ∈ Σ ∩ Z_{r}(p_{m}) , φ(x, t) > 0 and 0 ≤ t − t_{0} ≤ δ, the following estimate holds:

⟨∇(η v), μ⟩ < \frac{1}{2 ρ_{0}} v ((η v) − \frac{4 n α}{K}) .

Proof: We compute

(∗)  ⟨∇φ, μ⟩ ≤ |⟨μ, ω⟩| (\frac{r}{2 β} + 2 M_{r,δ}) − 2 ⟨F − p_{m}, μ⟩ .

By the choice of r, δ, we can apply Lemma 6.5 on the cylinder Z_{2 r}(p_{0}); this yields |u(x, t)| ≤ \frac{3}{4}r and thus

M_{r,δ} < r .

Now a lengthy but nevertheless straightforward calculation involving Lemma 6.3 and the Cauchy-Schwarz-inequality yields

⟨F − p_{m}, μ⟩ > \frac{3}{4} \frac{r}{K} , and |⟨μ, ω⟩|^{2} < 2(\frac{r}{ρ_{Σ}} )^{2} .

Substituting these expressions into equation (∗), we obtain

(∗∗)  ⟨∇φ, μ⟩ < −\frac{r}{K} .

Now we compute ⟨∇v, μ⟩ = −v^{2} A(ω^{T} , μ) ; using Assumption 6.1 (bounded curvature), this yields

(∗∗∗)  |⟨∇v, μ⟩| ≤ v^{2} |A| ∥ω^{T} ∥ ≤ v^{2} \frac{1}{2 ρ_{0}} .

Finally,

⟨∇(η v), μ⟩ = v ⟨∇η, μ⟩ + η ⟨∇v, μ⟩ = λ v (η + 1) ⟨∇φ, μ⟩ + η ⟨∇v, μ⟩ ,

and the result follows using equations (∗∗), (∗∗∗). □

Lemma 6.11 Let r > 0 , δ > 0 be as in Lemma 6.10 and α ≥ 2 , ε̂ ≤ 1. Then for any (x, t) ∈ M^{n} × [0, T] with F (x, t) ∈ Z_{r}(p_{m}) , 0 ≤ t − t_{0} ≤ δ, the following estimate holds:

(η v)(x, t) < sup_{Z_{r}, δ} η(., .) · max \{ \frac{α}{α−1} (8 β + 1), (1 + ε̂) sup_{Z_{r}} v(., t_{0}) \} .

Proof: Exactly as in the interior case, this estimate is valid at least until equality holds at a boundary point p = F (x_{2}, t_{2}) , x_{2} ∈ ∂M^{n} , 0 ≤ t_{2} − t_{0} ≤ δ. At this point, η v must have a maximum, hence we have

(∗)  ⟨∇(η v), μ⟩(x_{2}, t_{2}) ≥ 0 .

Now, by Lemma 6.4, sup_{Z_{r}} v(., t_{0}) < 2 ; furthermore a Taylor expansion of equation 6.3 yields sup_{Z_{r}, δ} η(., .) ≤ 4 n α\frac{r}{ρ_{0}} , and thus we have

(η v)(x_{2}, t_{2}) < 16 n α \frac{r}{ρ_{0}} .

Lemma 6.10 now implies ⟨∇(η v), μ⟩(p) < 0 in contradiction to (∗). □

We proceed as in the interior case by first estimating η from below on a smaller cylinder Z_{r_{1}}(p_{0}) ⊂ Z_{r}(p_{m}) and then Taylor expanding the quotient of the estimates for η from above (on Z_{r}(p_{m})) and from below (on Z_{r_{1}}(p_{0})). This can be done similarly as before, so we directly state the results:

Lemma 6.12 Let r ≤ \frac{1}{64} min\{ρ_{0} , \frac{ρ_{0}}{K^{2}} , \frac{ρ_{0}}{n α} , \frac{ρ_{Σ}}{K} , \frac{ρ_{Σ}^{2}}{K^{2} ρ_{0}} \} , δ ≤ r^{2} , r_{1} ≤ \frac{r}{K} and α ≥ 2 , K ≥ 8 , ε̂ ≤ 1 . Then for any (x, t) ∈ M^{n} × [0, T] with F (x, t) ∈ Z_{r_{1}}(p_{0}) and 0 ≤ t − t_{0} ≤ δ the following estimate holds:

v(x, t) < (1 + \frac{6}{K} + 4 n α \frac{r}{ρ_{0}} )·max \{ \frac{α}{α−1} (8 β + 1) , (1 + ε̂) sup_{Z_{r}} v(., t_{0}) \} . □

Theorem 6.13 Let p_{0} = F (x_{0}, t_{0}) ∈ ∂M_{t_{0}}. Then

∀ ε > 0 ∃ C_{i} > 0 , 1≤i≤6 , depending only on n, ε

such that

v(x, t) < 1 + ε ∀(x, t) with F (x, t) ∈ Z_{r_{1}}(p_{0}) , 0 ≤ t − t_{0} ≤ δ;

hereby r_{1} = min\{C_{1} ρ_{0} , C_{2} ρ_{Σ} , C_{3} \frac{ρ_{Σ}^{2}}{ρ_{0}} \} and δ = min\{C_{4} ρ_{0}^{2} , C_{5} ρ_{Σ}^{2} , C_{6} \frac{ρ_{Σ}^{4}}{ρ_{0}^{2}} \}. □

Thus again we have a gradient estimate on a time interval which is comparable to ρ_{Σ}^{2} , ρ_{0}^{2}. The Theorem immediately provides an estimate for the gradient of the SGR near the boundary point p_{0}; but, as Σ in general will be curved, this graph representation (which was defined with respect to Cartesian coordinates in IR^{n+1}) will be defined on a subset ∪_{t=t_{0}}^{t_{0}+δ} U_{t} ×{t} and not on a product structure U × [t_{0}, t_{0} +δ]. A graph representation on a product structure can be obtained by working with generalized Gaussian coordinates as in section 2. The gradient bound with respect to the latter graph representation is equivalent to estimating

ṽ(x, t) := ⟨ν, ξ∘F ⟩^{−1}(x, t)

from above. Now a connection between the above estimate for v = ⟨ν, ω⟩^{−1} and ṽ can easily be established by controlling ⟨ξ, ω⟩. This is a simple exercise of analytic geometry, and it turns out that Theorem 6.13 exactly carries over to ṽ. Thus we remark:

Remark 6.14 Theorems 6.7 and 6.13 yield a gradient bound for appropriate local graph representations w : Ur̂ × [t_{0}, t_{0}+δ] ⟶ IR of the M_{t} around a given point F (x_{0}, t_{0}) which is independent of F (x_{0}, t_{0}). Hence, by Lemma 6.5, we obtain a C^{1}-estimate for w on a domain Ur̂ × [t_{0}, t_{0} + δ̂], whose ”dimensions” have the natural scaling behaviour in ρ_{0}, i.e. r̂ ∼ ρ_{0} , δ̂ ∼ ρ_{0}^{2}.

6.4 Long Time Existence of Solutions

Proof of Theorem 1.3: According to Theorem 2.1, we have a solution to equation 1.2 on a maximal time interval [0, T_{C}). The only thing that remains to check is that, if

(∗)  sup\{ |A|^{2}(x, t) : x ∈ M^{n} , t ∈ [0, T^{C}) \} ≤ \frac{1}{2 n^{2}ρ_{0}} < ∞ ,

we can extend the solution to a larger time interval [0, T_{C} + δ). To this end, we start by defining the limit surface

F (x, T_{C}) := lim_{t→T_{C}} F (x, t) , M_{T_{C}} := F (M^{n}, T_{C}) ,

which is possible due to equation (∗) and Lemma 6.5. By the gradient estimates Theorems 6.7, 6.13, Remark 6.14 and (∗) we see that M_{T_{C}} can be covered by local graph representations of C^{1+α,(1+α)/2}-functions. As in chapter 2, each of these functions fulfils a uniformly parabolic differential equation on its domain of definition; on the boundary, in addition we have a transversal Neumann condition. Hence from linear parabolic theory we conclude that M_{T_{C}} must actually be a smooth hypersurface. Now we can apply the short time existence Theorem 2.1 with M_{T_{C}} as initial hypersurface and extend the given solution to a solution on [0, T_{C} + δ). □

7 The Neumann Boundary Condition

Having established a very general existence result, we now consider convex hypersurfaces Σ and strictly convex initial hypersurfaces M_{0}. We will show that a given solution of equation 1.2 develops a singularity within a finite time interval [0, T_{C}), and we are interested in the details of this process. To this end, we first analyze the given Neumann boundary condition

⟨ν, μ∘F ⟩(x, t) = 0 ∀(x, t) ∈ ∂M^{n} × [0, T] . (7.1)

Differentiating this equation with respect to time yields

Proposition 7.1 Let p = F (x, t) ∈ ∂M^{n} × (0, T]. Then

⟨∇H, μ⟩(p) = (H ·^{Σ}A(ν, ν))(p) .

Proof: \frac{d}{dt}ν = ∇H for t > 0, according to [13], Lemma 3.3. Moreover,

\frac{d}{dt}μ(F (x, t)) = dμ(\frac{d}{dt}F) = dμ(H) = −H dμ(ν) .

Hence, from equation 7.1 we obtain

0 = ⟨∇H, μ⟩ − H ⟨ν, dμ(ν)⟩ = ⟨∇H, μ⟩ − H^{Σ}A(ν, ν) . □

Next, we want to differentiate equation 7.1 with respect to a tangential space direction. This is best done using an orthonormal moving frame on M_{t} for a fixed t ≥ 0: We define orthonormal vector fields e_{1}, ..., e_{n+1} in IR^{n+1} such that for p ∈ Σ ∩ M_{t}

e_{1}(p), ..., e_{n−1}(p) ∈ T_{p}(Σ ∩ M_{t}) , e_{n}(p) = μ(p) , e_{n+1}(p) = ν(p) .

This means that the e_{α}, e_{i}, e_{I} are an orthonormal basis for the tangent bundles T IR^{n+1}, T M_{t}, T (Σ ∩M_{t}), respectively^{1}. We describe the covariant derivative in IR^{n+1} by

0∇_{e_{α}} e_{β} =: Γ^{σ}_{αβ}e_{σ} .

Note that the Γ^{σ}_{αβ} are not the usual Christoffel symbols which are deduced by using local coordinates; especially they are not symmetric with respect to the lower indices. Instead, using the relations between covariant derivatives and the second fundamental form:

(7.2)  \begin{matrix}^{0}∇_{e_{i}} e_{j} = ∇_{e_{i}} e_{j} − A(e_{i}, e_{j})ν , & ^{0}∇_{e_{i}} ν = A(e_{i}, e_{p})e_{p} , \\ ^{0}∇_{eî} eĵ = ^{Σ}∇_{eî} eĵ −^{Σ}A(eî, eĵ)μ , & ^{0}∇_{eî} μ = ^{Σ}A(eî, ep̂) ep̂ , \end{matrix}

we obtain the following equations^{2}:

Γ^{σ}_{αβ} = −Γ^{β}_{ασ} , Γ^{ν}_{ij} = −A(e_{i}, e_{j}) , Γ^{μ} î ; ĵ = −^{Σ}A(eî, eĵ) . (7.3)

Finally, a straightforward calculation yields for the covariant derivative of symmetric tensors T, T_{Σ} on M_{t}, Σ, respectively:

(7.4)

(∇_{e_{k}}T)(e_{i}, e_{j}) = d(T (e_{i}, e_{j}))(e_{k}) −Γ^{p}_{ki} T (e_{p}, e_{j}) −Γ^{q}_{kj} T (e_{i}, e_{q}) ,
(^{Σ}∇_{ek̂}T_{Σ})(eî, eĵ) = d(T_{Σ}(eî, eĵ))(ek̂)−Γp̂_{k̂î}T_{Σ}(ep̂, eĵ)−Γq̂_{k̂ĵ}T_{Σ}(eî, eq̂) .

Using these relations, we can differentiate the Neumann boundary condition in a tangential space direction: From equations 7.1 and 7.2, we derive

0 = ⟨^{0}∇_{e_{I}} ν, μ⟩ + ⟨ν,^{0}∇_{e_{I}} μ⟩ = A(e_{I}, μ) +^{Σ}A(e_{I}, ν) .

Since e_{1}(p), ..., e_{n−1}(p) span the tangent space T_{p}(Σ ∩ M_{t}), we obtain

Proposition 7.2 Let p ∈ Σ ∩ M_{t} , v ∈ T_{p}M_{t} and w := v − ⟨v, μ⟩μ ∈ T_{p}(M_{t} ∩ Σ) be the projection of v onto T_{p}Σ. Then

\begin{aligned}A(w, μ) & = & −^{Σ}A(w, ν) and \\ A(v, μ) & = & −^{Σ}A(w, ν) + ⟨v, μ⟩A(μ, μ) . \end{aligned} □

Moreover, we have a relation between the covariant derivatives of the second fundamental forms of M_{t} and Σ in tangential directions:

1Recall the range of the various summation indices !

2Here and in the sequel we replace the indices _{n},_{n+1} by the somewhat more suggestive notation _{μ},_{ν}

Proposition 7.3 Let p ∈ Σ ∩ M_{t} and v, w ∈ T_{p}(Σ ∩ M_{t}). Then

(∇v A)(μ, w) = −(^{Σ}∇v^{Σ}A)(ν, w)
−A(v, e_{P})^{Σ}A(e_{P} , w) − A(w, e_{P})^{Σ}A(e_{P} , v)
+ A(μ, μ)^{Σ}A(v, w) + A(v, w)^{Σ}A(ν, ν) .

Proof: Using equations 7.2 through 7.4 with T = A, we obtain

(∇_{e_{I}} A)(μ, e_{J}) = d(A(μ, e_{J}))(e_{I}) −^{Σ}A(e_{I}, e_{P}) A(e_{P} , e_{J})
−Γ^{Q}_{IJ} A(μ, e_{Q}) +^{Σ}A(e_{I}, e_{J}) A(μ, μ) .

Analogously, for T_{Σ} = ^{Σ}A:

(^{Σ}∇_{e_{I}}^{Σ}A)(ν, e_{J}) = d(^{Σ}A(ν, e_{J}))(e_{I}) − A(e_{I}, e_{P})^{Σ}A(e_{P} , e_{J})
−Γ^{Q}_{IJ} ^{Σ}A(ν, e_{Q}) + A(e_{I}, e_{J})^{Σ}A(ν, ν) .

Now, according to Proposition 7.2, we have A(μ, e_{J}) +^{Σ}A(ν, e_{J}) ≡ 0 on the boundary and hence d (A(μ, e_{J}) +^{Σ}A(ν, e_{J})) (e_{I}) = 0 . Thus we obtain the result by summing the above equations. □

We finish this section by summarizing the results. If we define

h_{ij}(x, t) := A(e_{i}, e_{j})(x, t) , ∇_{k} h_{ij}(x, t) := (∇_{e_{k}} A)(e_{i}, e_{j})(x, t)

etc., we have:

Theorem 7.4 On the boundary Σ ∩ M_{t}, the components of the second fundamental forms of M_{t} and Σ are related by the equations

(i) h_{μI} = −h_{νI}^{Σ} , (ii) ∇_{I} h_{μJ} = −^{Σ}∇_{I} h_{νJ}^{Σ} − h_{IP} h_{P J}^{Σ} − h_{JP} h_{P I}^{Σ} + h_{μμ}h_{IJ}^{Σ} + h_{IJ}h_{νν}^{Σ} , (iii) ∇_{μ} H = H h_{νν}^{Σ} . □

Note that the equations (i), (ii) are valid for t ≥ 0, whereas equation (iii) requires t > 0.

Remark 7.5 Given a point p = F (x, t) ∈ Σ ∩ M_{t}, we can introduce local coordinates around p such that the vector fields e_{I}(p) := \frac{∂}{∂x^{I}} F (x, t), e_{n}(p) := \frac{∂}{∂x^{n}} F (x, t) = μ(p) and e_{n+1} := ν(x, t) are orthonormal at p; this relates the formalisms of orthonormal moving frames and local coordinate systems.

8 Curvature Bounds

This section is devoted to the study of the mean curvature of the hypersurfaces M_{t} of a given solution to equation 1.2. The evolution equation for H is

\frac{∂}{∂t}H = ΔH + |A|^{2} H , (8.1)

Δ being the Laplace-Beltrami-operator of M^{n} with respect to the metric g (cf. [13], Corollary 3.5).

8.1 Estimates from Below

Theorem 8.1 Let H(., 0) ≥ 0. Then H ≥ 0 on M^{n} × [0, T]; moreover, if H(., 0) ≢ 0, we have H > 0 ∀t > 0.

Proof: According to Proposition 7.1, we have ⟨∇H, μ⟩(p) = (H·^{Σ}A(ν, ν))(p). Thus, regarding the evolution equation 8.1, the conditions of the weak maximum principle Theorem 3.1 are fulfilled for H; hence H ≥ 0 is preserved. The strong maximum principle Theorem 3.2 now yields the result. □

Theorem 8.2 Let Σ be convex with respect to the exterior normal field μ, and let H(., 0) ≥ C_{0} > 0. Then H ≥ C_{0} for all t ∈ [0, T_{C}), and the maximal time interval [0, T_{C}) for the solution is bounded by T_{C} ≤ t_{0} := \frac{n}{2 C_{0}^{2}} .

Proof: The solution u(t) of the ordinary initial value problem

\frac{d}{dt}u = \frac{1}{n} u^{3} , u(0) = C_{0}

is given by u(t) = C_{0} · (1 − 2 C_{0}^{2} t/n)^{−1/2}. Now, using the simple algebraic identity |A|^{2} ≥ \frac{1}{n}H^{2}, we see that the function φ(x, t) := H(x, t)−u(t) fulfils the following differential equation:

(\frac{∂}{∂t} − Δ)φ ≥ \frac{1}{n}(H^{3} − u^{3}) = \frac{1}{n}(H^{2} + H u + u^{2})(H − u) =: c(x, t) φ .

Moreover, by the convexity of Σ and since u > 0, we have on the boundary

⟨∇φ, μ⟩ = ^{Σ}A(ν, ν) H ≥ ^{Σ}A(ν, ν) H −^{Σ}A(ν, ν) u = ^{Σ}A(ν, ν) φ .

Thus we can apply the maximum principles Theorems 3.1, 3.2 to φ and obtain the result. □

It can easily be seen that the above estimate on the maximal existence interval is sharp for convex hypersurfaces without boundary, since the sphere S_{R}^{n} homothetically shrinks to a point in T_{C} = \frac{R^{2}}{2 n} = \frac{n}{2 C_{0}^{2}} = t_{0}. Thus, t_{0} can also be reached in the boundary-case: Let G be a cone, Σ = ∂G and M_{0} part of a sphere S_{R}^{n} such that the center of the sphere coincides with the vertex of the cone. Note that the boundary condition is fulfilled as M_{t} shrinks homothetically to the vertex of the cone. Now, the natural question is whether this is the only setup with maximal existene time of the solution. We prove:

Theorem 8.3 Let G ⊂ IR^{n+1} be a convex domain with smooth boundary Σ. Let F be the solution of equation 1.2, where M_{0} has strictly positive mean curvature. Then T_{C} = t_{0} if and only if G is a cone with vertex p and M_{0} ⊂ S_{R}(p).

Proof: We define U(t) := min_{M^{n}} H(x, t) and obtain from the proof of Theorem 8.2

U(t) ≥ _{\frac{C_{0}}{\sqrt{1−2 C_{0}^{2} t/n}}}

Now, using Theorem 8.2, we conclude that in order to obtain T_{C} = t_{0}, actually equality must hold; thus U(t) fulfils the ordinary initial value problem

\frac{d}{dt}U = \frac{1}{n} U^{3} , U(0) = C_{0} .

Now we consider the function φ(x, t) := H(x, t) − U(t). Since for every t due to the compactness of M^{n} there exists a zero of φ(., t), by the strong maximum principle we conclude φ ≡ 0 ∀t; thus H(., t) ≡ U(t) on M_{t}. This in turn implies ∇H ≡ 0 and Proposition 7.1 yields ^{Σ}A(ν, ν) ≡ 0, since H ≥ C_{0} > 0. Furthermore, from ΔH ≡ 0 and the evolution equations for H and U we conclude |A|^{2} = \frac{1}{n}H^{2} and thereby that the M_{t} are spheres. Finally the equation \frac{∂}{∂t}ν = ∇H ([13], Lemma 3.3) implies ν(., t) = const(t), and so these spheres must be concentric; the result follows. □

We remark that this proof equally holds for the boundaryless case: Here the only way to obtain the maximal existence interval of a solution which starts with strictly positive mean curvature is M_{0} being a sphere.

8.2 Estimate from Above

As we have seen, it is not hard to establish lower bounds on H under reasonably weak conditions. In contrast, it is much harder to control the increasing of H from above; this can be done if the M_{t} are guaranteed to be convex. Now convexity of the M_{t} is preserved in the special case of Σ being umbilic, cf. Theorem 9.3. Thus in the sequel we will require Σ to be the boundary of a sphere or a hyperplane, but we emphasize that the following method also applies to arbitrary convex hypersurfaces Σ, if by any means we can guarantee that convexity of the M_{t} is preserved.

In this section, we apply a technique due to Tso [24] who studied closed hypersurfaces moving under the Gauss-Kronecker-flow.

Assumption 8.4 For the remainder of this section we let G = B_{R}^{n+1} or G be a half space and Σ = ∂G. Furthermore, F is the solution of equation 1.2 with M_{0} ⊂ Ḡ being an imbedded, convex hypersurface.

Thus all M_{t} will be imbedded hypersurfaces due to Corollary 4.5, and since H(., 0) ≥ 0, this will equally hold for t > 0 (Theorem 8.1). Finally, convexity of M_{t} for t > 0 follows from Theorem 9.3.

We start our considerations by establishing evolution equations for some auxiliary functions: First, there exists δ_{0} > 0 and a neighbourhood U := {p ∈ IR^{n+1} : dist_{n+1}(p, Σ) < δ_{0}} of Σ such that the distance function

g : U ⟶ IR , g(p) := dist_{n+1}(p, Σ)

is well defined on U and smooth on U\Σ and fulfils the estimates

∥Dg∥ ≤ 1 , |D^{2}g| < C_{1} , (8.2)

where C_{1} = C_{1}(^{Σ}A) > 0 depends only on the curvature of Σ. Moreover, if p ∈ Σ, we have

⟨Dg, μ⟩(p) = −1 . (8.3)

Lemma 8.5 (The distance function) The function d : M^{n} × [0, T] ⟶ IR, d(x, t) := (g∘F)(x, t) fulfils the evolution inequality

|(\frac{∂}{∂t} − Δ)d(x, t)| ≤ n C_{2} ,

where the constant C_{2} > 0 depends only on ^{Σ}A. Furthermore,

⟨∇d, μ∘F ⟩ = −1 on ∂M^{n}×[0, T] , and ∥∇d∥ ≤ 1 on U .

Proof: A straightforward calculation yields

(\frac{∂}{∂t} − Δ)d(x, t) = −g^{ij} · (D^{2}g∘F)(\frac{∂}{∂x^{i}} F , \frac{∂}{∂x^{j}} F)(x, t) ;

thus the evolution inequality for d follows from equation 8.2. Analogously, the boundary behaviour is a consequence of

⟨∇d, μ∘F ⟩(x, t) = ⟨Dg∘F , μ∘F ⟩(x, t) .

Finally, ∥∇d∥^{2} = ∥Dg∥^{2} on U; again, equation 8.2 yields the result. □

We remark that for convex hypersurfaces Σ we have (\frac{∂}{∂t} − Δ)d ≥ 0; unfortunately, this turns out to be the wrong sign for the desired estimates.

Lemma 8.6 (The support function) The function s : M^{n} × [0, T] ⟶ IR, s(x, t) := ⟨ν, F ⟩(x, t) fulfils

(\frac{∂}{∂t} − Δ)s(x, t) = −|A|^{2} + 2 s H ∀t > 0 ,
⟨∇s, μ∘F ⟩ = A(F^{T}, μ∘F) on the boundary ;

hereby F^{T} is the projection of F onto the tangent hyperplane of M_{t}.

Proof: The evolution equation was proved by Ecker and Huisken [6]. The boundary condition is obtained by a straightforward calculation. □

Now we let p ∈ Ḡ be the origin of a coordinate system in IR^{n+1}, and we choose σ > 0 and T_{σ} > 0 such that s(., t) ≥ 2 σ on [0, T_{σ}]. (This is possible due to the convexity of the imbeddings M_{t}.) We then consider the function

f : M^{n} × [0, T_{σ}] ⟶ IR , f(x, t) := H \frac{ψ ∘ d}{s − σ} (x, t) ,

ψ : IR_{+} ⟶ IR_{+} being a C^{2}-function which will be chosen later.

Lemma 8.7 For t > 0, we have

(\frac{∂}{∂t} − Δ)f ≤ − \frac{σ (s−σ)}{n (ψ∘d)^{2}} f^{3} + ( \frac{2}{ψ∘d} + 2 ∥F ∥ |\frac{ψ′}{ψ^{2}} ∘d| ) f^{2}
+ ( n C_{2} |\frac{ψ′}{ψ} ∘d| + |\frac{2 ψ^{′2}−ψ ψ′′}{ψ^{2}} ∘d| ) f
+ ⟨∇Q, ∇f⟩ ,
⟨∇f, μ⟩ ≤ f (C_{Σ} − \frac{ψ′}{ψ} (0) + \frac{1}{s−σ} C_{Σ} ∥F ∥) on the boundary ;

hereby Q := 2 log(\frac{s−σ}{ψ∘d}) and C_{Σ} := ∥^{Σ}A∥_{∞} = sup\{ |^{Σ}A|(p) : p ∈ Σ \}.

Proof: Using equation 8.1 and Lemmata 8.5, 8.6, we obtain by a straightforward calculation

(\frac{∂}{∂t} − Δ)f ≤ f \frac{1}{s−σ} (−σ |A|^{2} + 2 H) + f |\frac{ψ′}{ψ} ∘d| n C_{2}
− f \frac{2}{s−σ} (\frac{ψ′}{ψ} ∘d) ⟨∇s, ∇d⟩ + f (\frac{2 ψ^{′2}−ψ ψ′′}{ψ^{2}} ∘d) ∥∇d∥^{2}
+ 2 \frac{ψ∘d}{s−σ} ⟨∇\frac{s−σ}{ψ∘d} , ∇f⟩ .

Moreover, we compute ⟨∇s, ∇d⟩ = A(F^{T} , ∇d⟩ , and thereby

|⟨∇s, ∇d⟩| ≤ |A| ∥F ∥ ∥∇d∥ .

Putting this into the above evolution equation for f and using the inequality \frac{1}{n}H^{2} ≤ |A|^{2} ≤ H^{2} (which is valid since the M_{t} are convex hypersurfaces), we obtain

(\frac{∂}{∂t} − Δ)f ≤ f \frac{1}{s−σ} (−\frac{σ}{n} H^{2} + 2 H) + f |\frac{ψ′}{ψ} ∘d| n C_{2}
+ f \frac{2}{s−σ} |\frac{ψ′}{ψ} ∘d| H ∥F ∥ + f |\frac{2 ψ^{′2}−ψ ψ′′}{ψ^{2}} ∘d|
+ ⟨∇Q, ∇f⟩ .

Finally, using H = f \frac{s−σ}{ψ∘d} , the evolution equation follows.

On the boundary, we have

∇f = \frac{ψ(0)}{s−σ} ∇H + H (\frac{ψ′(0)}{s−σ} ∇d − \frac{ψ(0)}{(s−σ)^{2}} ∇s) .

By Proposition 7.1 and Lemmata 8.5, 8.6, we arrive at

⟨∇f, μ⟩ = \frac{ψ(0)}{s−σ} ^{Σ}A(ν, ν) H − H (\frac{ψ′(0)}{s−σ} + \frac{ψ(0)}{(s−σ)^{2}} A(F^{T} , μ)) .

Corollary 7.2 gives

A(F^{T}, μ) = ⟨F^{T}, μ⟩ A(μ, μ) −^{Σ}A(F^{T} −⟨F^{T},μ⟩μ , ν) ;

here, A(μ, μ) ≥ 0, as the M_{t} are convex. Furthermore, Σ bounds a convex domain, and thus the boundary condition ⟨μ, ν⟩ = 0 yields ^{3} ⟨F^{T}, μ⟩ = ⟨F, μ⟩ ≥ 0 . Hence

A(F^{T}, μ) ≥ −C_{Σ} ∥F^{T} − ⟨F^{T}, μ⟩μ∥ ≥ −C_{Σ} ∥F ∥ ,

3Recall that the origin of our coordinate system is in Ḡ!

and putting this into the above expression for ⟨∇f, μ⟩ leads to the stated boundary condition. □

Before stating the Theorem, we observe that, with respect to the choice of our coordinates, the hypersurfaces M_{t} ”shrink” as t increases; more precisely, we have

sup{∥F (x, t)∥ : (x, t) ∈ M^{n}×[0, T_{σ}]} = sup{∥F (x, 0)∥ : x ∈ M^{n}} . (8.4)

This follows from equation 1.2 by calculating \frac{d}{dt}∥F ∥^{2}(x, t) = −2 H s ≤ 0.

We now show that the mean curvature H is bounded from above as long as the support function s is strictly positive; geometrically this latter condition means that the convex body bounded by M_{t} and Σ contains a small ball.

Theorem 8.8 (An upper bound on the growth of H) Let G ⊂ IR^{n+1} be a ball or a half space, Σ := ∂G, C_{Σ} := ∥^{Σ}A∥_{∞} and M_{0} ⊂ Ḡ a convex imbedded hypersurface. Let F : M^{n}×[0, T] ⟶ IR^{n+1} be a solution of equation 1.2 with F (M^{n}, 0) = M_{0}. Finally, choose ^{4} σ ∈ (0 , sup_{M^{n}} ∥F (., 0)∥ · min{C_{Σ} δ_{0}, 1}) and T_{σ} ∈ (0, T] such that s(., t) ≥ 2 σ ∀t ∈ [0, T_{σ}]. Then

H(x, t) ≤ max\{ sup_{M^{n}} H(., 0) , C_{0} \frac{n}{σ^{3}} \} ∀ (x, t) ∈ M^{n} × [0, T_{σ}] ,

where C_{0} is a positive constant depending only on the curvature bound C_{Σ} and the initial imbedding M_{0}.

Proof: Define k_{0} := sup_{M^{n}} ∥F (., 0)∥ , δ := \frac{σ}{C_{Σ} k_{0}} < δ_{0} , C := \frac{1}{4 δ^{3}} and

ψ(z) := \{ \begin{matrix}C (z − δ)^{3} + 1/2 , & if z ≤ δ , \\ 1/2 , & otherwise. \end{matrix}

Then ψ ∈ C^{2}(IR_{+}) and ψ ∈ [1/4, 1/2] . We now consider the function f of Lemma 8.7:

(i) Using equation 8.4 and σ < k_{0}, the boundary behaviour is

⟨∇f, μ⟩ ≤ f (C_{Σ} − \frac{3 C δ^{2}}{1/2−C δ^{3}} + C_{Σ} \frac{k_{0}}{σ} ) < f (C_{Σ} \frac{2 k_{0}}{σ} − \frac{3}{δ}) < 0 ;

hence a maximum of f cannot occur on the boundary.

(ii) For any point (x, t) ∈ [0, T_{σ}] with dist_{n+1}(F (x, t), Σ) > δ, we have ψ ≡ 1/2 and thus, using s ≥ 2 σ on [0, T_{σ}] , the evolution equation for f yields

(\frac{∂}{∂t} − Δ)f ≤ 4 f^{2} (−\frac{σ^{2}}{n} f + 1) + ⟨∇Q, ∇f⟩ .

Hence f cannot develop a new maximum at (x, t) with f ≥ \frac{n}{σ^{2}} .

4δ_{0} is as defined at the beginning of this subsection.

(iii) Now let dist_{n+1}(F (x, t), Σ) ≤ δ . Here we have 2 σ ≤ s ≤ ∥F ∥ ≤ k_{0} and 1/4 ≤ ψ ≤ 1/2, and thereby

(\frac{∂}{∂t} − Δ)f ≤ −\frac{4 σ^{2}}{n} f^{3} + (8 + 96 C δ^{2} k_{0}) f^{2}
+ 12 C δ (n δ C_{2} + 16 C δ^{3} + 4) f
+ ⟨∇Q, ∇f⟩ .

By Young’s inequality,

(8 + 96 C δ^{2} k_{0}) f ≤ \frac{2 σ^{2}}{n} f^{2} + 8 \frac{n}{σ^{2}} (1 + 12 C δ^{2} k_{0})^{2} ,

and using δ = \frac{σ}{C_{Σ} k_{0}} , σ < k_{0} and C δ^{3} = 1/4, we arrive at

(\frac{∂}{∂t} − Δ)f ≤ f (−\frac{2 σ^{2}}{n} f^{2} + 2 n C_{3}^{2} \frac{1}{σ^{4}} ) + ⟨∇Q, ∇f⟩

with a constant C_{3} > 0 depending only on M_{0} and C_{Σ}. Thus, the occurence of new maxima with f ≥ C_{3} \frac{n}{σ^{3}} can be ruled out.

(iv) From (i)-(iii) we conclude that H cannot develop a new maximum

with

H ≥ \frac{n C_{3}}{σ^{3}} 4 k_{0} =: C_{0} \frac{n}{σ^{3}}

within the time interval [0, T_{σ}]. □

Remark 8.9 The above proof and thus the Theorem hold for arbitrary convex domains G, if we can guarantee that convexity is preserved for the hypersurfaces M_{t}.

Remark 8.10 Since for convex hypersurfaces |A|^{2} ≤ H^{2}, by controlling the mean curvature we automatically control the entire second fundamental form of the M_{t}.

9 Convexity and Pinching

Here we study the properties of the second fundamental form of a given solution to the mean curvature flow equation 1.2: We show that (under suitable conditions on Σ) convexity and a pinching estimate are preserved. Hereby we require Σ to be an umbilic hypersurface in IR^{n+1}, as in this case by the results of section 7 we can completely control the behaviour of the various tensors on the boundary. We will work with the maximum principle Theorem 3.3 and begin by summarizing some evolution equations from [13]:

(9.1)  \frac{∂}{∂t}g_{ij} = −2 H h_{ij} ,
(9.2)  \frac{∂}{∂t}h_{ij} = Δh_{ij} − 2 H h_{ik} g^{kl} h_{lj} + |A|^{2} h_{ij} ,
(9.3)  \frac{∂}{∂t}|A|^{2} = Δ|A|^{2} − 2 |∇A|^{2} + 2 |A|^{4} .

Assumption 9.1 Throughout this section, let Σ be the boundary of a ball B_{1/α}^{n+1} or a hyperplane in IR^{n+1}. In the former case, let M_{0} ⊂ B̄_{1/α}^{n+1}, thereby Σ being convex.

Thus we have

ΣA(v, w)(p) = α^{Σ}g(v, w)(p) ∀p ∈ Σ , ∀v, w ∈ T_{p}Σ ,

or, equivalently, for the components with respect to a parametrisation,

hΣ ; î ; ĵ(p) = α^{Σ}gî ; ĵ(p) ∀p ∈ Σ , ∀î, ĵ .

Thereby α ≥ 0 due to the convexity of Σ.

Theorem 9.2 Let Σ be umbilic, p ∈ Σ ∩ M_{t}, and choose local coordinates around p as discussed in Remark 7.5. Then the following relations hold:

\begin{aligned}(i) & h_{μI} & = & 0 , \\ (ii′) & ∇_{μ} h_{IJ} & = & α (−h_{IJ} + h_{μμ} δ_{IJ}) , \\ (ii′′) & ∇_{μ} h_{μμ} & = & α (2 H − n h_{μμ}) , \\ (iii) & ∇_{μ} H & = & α H . \end{aligned}

Proof: (i) follows immediately from h_{νI}^{Σ} = α δ_{νI} = 0 and Theorem 7.4,(i). Moreover, by Ricci’s Lemma, ^{Σ}∇î^{Σ}g_{p̂q̂} ≡ 0; hence ^{Σ}∇_{I} h_{νJ}^{Σ} = α^{Σ}∇_{I} ^{Σ}g_{νJ} = 0. Theorem 7.4,(ii) thus yields ∇_{I} h_{μJ} = α (−h_{IJ} + h_{μμ} δ_{IJ}), and (ii)′ is a consequence of the Codazzi equations ∇_{I} h_{μJ} = ∇_{μ} h_{IJ}. (iii) follows from h_{νν}^{Σ} = α δ_{νν} = α, and combining (ii)′ and Theorem 7.4,(iii) yields (ii). □

Thus if Σ is umbilic, the normal derivative of the second fundamental form of M_{t} can be expressed essentially in terms of the curvature operator itself. Furthermore, in this case the second fundamental form A decomposes on Σ, cf. Lemma 3.4.

9.1 Convexity

Theorem 9.3 Let Σ be convex and umbilic. Then convexity is preserved for any solution of the mean curvature flow equation 1.2.

Proof: We want to apply Theorem 3.3. To this end, we set M_{ij} := h_{ij} , N_{ij} := −2 H h_{ik} g^{kl} h_{lj} + |A|^{2} h_{ij} , u^{k} = 0 and observe that the required evolution equation and the zero eigenvector condition are fulfilled. Furthermore, h_{ij}(., 0) ≥ 0 by assumption, and thus in order to apply Theorem 3.3 we only have to check the boundary condition. Let p := F (x_{0}, t_{0}) ∈ Σ and choose local coordinates around p as in Remark 7.5. As A decomposes on Σ (cf. Theorem 9.2,(i)), we can use Lemma 3.4:

(iv′) If v := v^{I} ∂_{I} is a minimal eigenvector of h_{ij} with h_{ij}v^{i}v^{j} = h_{IJ}v^{I}v^{J} = −δ∥v∥^{2} ≤ 0, we compute from Theorem 9.2,(ii)′ ∇_{μ} h_{ij}v^{i}v^{j} = α (δ +

h_{μμ}) ∥v∥^{2}. As v is minimal, h_{μμ} ≥ −δ and thus ∇_{μ} h_{ij} v^{i} v^{j} ≥ 0.

(iv′′) Now let h_{μμ} ≤ 0 be minimal. Then Theorem 9.2,(ii′′) yields ∇_{μ} h_{μμ} ≥ 2 α H. By assumption, H(., 0) ≥ 0, and by Theorem 8.1, this is preserved; thus ∇_{μ} h_{μμ} ≥ 0.

Hence the result follows from Theorem 3.3 and Lemma 3.4. □

Proposition 9.4 Let Σ be convex and umbilic and C ≤ \frac{2}{n} min_{M^{n}} H(., 0). Then the estimate

h_{ij} ≥ C g_{ij}

is preserved for any solution of equation 1.2.

Proof: Let M_{ij} := h_{ij} −C g_{ij}. A straightforward calculation using equations 9.1, 9.2 yields

(\frac{∂}{∂t} − Δ)M_{ij} ≥ −2 H M_{ik} g^{kl} M_{lj} + (|A|^{2} − 2 CH) M_{ij} ,

thus a ”suitable” evolution equation (with respect to Theoren 3.3) and the zero eigenvector condition are fulfilled. By assumption, M_{ij}(., 0) ≥ 0; thus again the only thing to check is the boundary condition of Lemma 3.4. Using Theorem 9.2 and Ricci’s Lemma, we compute

∇_{μ} M_{IJ} = α (−M_{IJ} + M_{μμ}) ,
∇_{μ} M_{μμ} = α (2 H − n M_{μμ} − n C) .

Now, if v = v^{I} ∂_{I} is a minimal eigenvector with M_{IJ} v^{I} v^{J} = −δ ∥v∥^{2} ≤ 0, we have M_{μμ} ≥ −δ and thus ∇_{μ} M_{IJ} v^{I} v^{J} = α (δ + M_{μμ})∥v∥^{2} ≥ 0. If, on the other hand, M_{μμ} ≤ 0 is minimal, we compute ∇_{μ} M_{μμ} ≥ 2 α (H − min_{M^{n}} H(., 0)). As Σ is convex, we have H ≥ min_{M^{n}} H(., 0) by Theorem 8.2; thus ∇_{μ} M_{μμ} ≥ 0.

Hence we can apply Theorem 3.3 and Lemma 3.4 and obtain the result. □

9.2 Pinching

Now we want to establish various pinching estimates. Huisken [13] in the boundaryless case proved that for any 0 < ε ≤ 1/n < κ < 1 the estimate

ε H g_{ij} ≤ h_{ij} ≤ κ H g_{ij}

is preserved. Note that this estimate is optimal as ε and κ cannot exceed the above stated range for any convex hypersurface. Now, in our case, such an optimal estimate cannot hold: Let M_{0} be a subset of S_{R}^{n} and Σ = S_{1/α}^{n} such that the Neumann boundary condition is fulfilled. Then h_{ij} = \frac{1}{n}H g_{ij}, i.e. ε = 1/n; but an elementary geometric consideration shows that under the mean curvature flow M_{t} will not be a sphere for t > 0; thus the optimal pinching is not preserved.

Theorem 9.5 Let Σ be convex and umbilic, and let κ ≥ \frac{2}{n+1}. Then equation 1.2 preserves the pinching condition

h_{ij} ≤ κ H g_{ij} .

Proof:

Let M_{ij} := κ H g_{ij} − h_{ij}. By a straightforward calculation,

(\frac{∂}{∂t} − Δ)M_{ij} = 2 H M_{ik} g^{kl} M_{lj} + (|A|^{2} − 2 κ H^{2}) M_{ij} ,

i.e. the evolution equation and the zero eigenvector condition are fulfilled. By assumption M_{ij}(., 0) ≥ 0, and thus it remains to check the boundary condition. Here, Theorem 9.2 yields

∇_{μ} M_{IJ} = α (κ H δ_{IJ} − M_{IJ} + M_{μμ} δ_{IJ}) ,
∇_{μ} M_{μμ} = α (−n M_{μμ} + ((n+1)κ − 2) H) .

Thus, if v = v^{I} ∂_{I} is a minimal eigenvector with M_{IJ}v^{I}v^{J} = −δ ∥v∥^{2} ≤ 0, we have ∇_{μ} M_{IJ}v^{I}v^{J} = α (κ H + δ + M_{μμ}) ∥v∥^{2} ≥ 0, as H ≥ 0 is preserved. On the other hand, if M_{μμ} ≤ 0 is minimal, we obtain ∇_{μ} M_{μμ} ≥ α ((n+1)κ − 2) H ≥ 0, as κ ≥ \frac{2}{n+1}. Thus Theorem 3.3 and Lemma 3.4 yield the result. □

Now we would like to proceed as in [13] and establish a pinching estimate from below using the function M_{ij} = h_{ij} −ε H g_{ij}; unfortunately, the normal derivative of this function at the boundary is ∇_{μ} M_{IJ}v^{I}v^{J} ≥ −α ε H ∥v∥^{2}, if v is a minimal tangential eigenvector; this is not sufficient to guarantee for the requirements of Lemma 3.4. It turns out that the negative term occurs from the boundary behaviour of H - thus we must replace H by a function with a ”better” boundary behaviour.

Lemma 9.6 Let Σ be convex and umbilic. Furthermore, let M_{0} be strictly convex. Then

(\frac{∂}{∂t} − Δ)|A| ≤ |A|^{3} ,

and on the boundary (with respect to the ”usual” coordinates)

∇_{μ} |A| = α |A|^{−1} (3 H h_{μμ} − |A|^{2} − n h_{μμ}^{2}) .

Proof: First note that |A|^{2} > 0 is preserved (Theorem 9.4). Thus |A| is a smooth function, and we compute

(\frac{∂}{∂t} − Δ)|A| = \frac{1}{|A|} (∥∇|A|∥^{2} − |∇A|^{2}) + |A|^{3} .

The evolution equation then follows from the Cauchy-Schwarz inequality ∥∇|A|∥^{2} ≤ |∇A|^{2}.

Furthermore, using Theorem 9.2, a simple calculation yields

∇_{μ} |A|^{2} = 2 α (3 H h_{μμ} − |A|^{2} − n h_{μμ}^{2})

on Σ, and the result follows from ∇_{μ} |A| = \frac{1}{2 |A|} ∇_{μ} |A|^{2}. □

Theorem 9.7 Let Σ be convex and umbilic, and let M_{0} be strictly convex. Then for ε ≤ min\{\frac{1}{3 \sqrt{n}}, \frac{2}{n}\}, equation 1.2 preserves the pinching estimate

h_{ij} ≥ ε |A| g_{ij} .

Proof: We define M_{ij} := h_{ij} −ε |A| g_{ij} and compute, using the above Lemma

(\frac{∂}{∂t} − Δ)M_{ij} ≥ −2 H M_{ik} g^{kl} M_{lj} + (|A|^{2} − 2 ε H |A|) M_{ij} ,

and at a boundary point

∇_{μ} M_{IJ} = α (−M_{IJ} + M_{μμ} (1 − 3 ε H |A|^{−1}) δ_{IJ}
+ ε (−3 ε H + |A| + n |A|^{−1} h_{μμ}^{2}) δ_{IJ}) ,
∇_{μ} M_{μμ} = α (−(n + 3 ε H |A|^{−1}) M_{μμ} + (2 − 3 ε^{2}) H
− ε (n−1) |A| + n ε |A|^{−1} h_{μμ}^{2}) .

As convexity is preserved, we have \frac{1}{n}H^{2} ≤ |A|^{2} ≤ H^{2}; furthermore, by assumption, ε ≤ \frac{1}{3 \sqrt{n}} and thus λ := 1 − 3 ε H |A|^{−1} ∈ [0, 1].

Now, if v = v^{I} ∂_{I} is a minimal eigenvector with M_{IJ} v^{I} v^{J} =−δ ∥v∥^{2} ≤ 0,

∇_{μ} M_{ij}v^{i}v^{j} ≥ α (δ + M_{μμ} λ) ∥v∥^{2} ;

and M_{μμ} ≥ −δ (by minimality of −δ) and 0 ≤ λ ≤ 1 yield ∇_{μ} M_{ij}v^{i}v^{j} ≥ 0.

If M_{μμ} = −δ ≤ 0 is minimal, we have

∇_{μ} M_{μμ} ≥ α |A| (2 − 3 ε^{2} − ε (n−1)) ≥ 0 ,

by the assumption on ε.

Again the result follows from Theorem 3.3 and Lemma 3.4. □

Corollary 9.8 Let Σ be convex and umbilic, M_{0} strictly convex and ε ≤ min\{\frac{1}{3\sqrt{n}}, \frac{2}{n}\}. Then the estimate h_{ij}(., 0) ≥ ε (H g_{ij})(., 0) implies

h_{ij} ≥ \frac{ε}{\sqrt{n}}H g_{ij} on M^{n} × [0, T] .

Proof: This follows directly from Theorem 9.7 and the estimate \frac{1}{n}H^{2} ≤ |A|^{2} ≤ H^{2} for convex hypersurfaces. □

10 Convergence

We finally turn to the study of the singularity which develops as t → T_{C}. The main results have been stated in Theorem 1.4 and Proposition 1.5; thus it essentially remains to give the proofs. As before, we let Σ be the boundary of a ball or a half space in IR^{n+1}; however, in this section we will not use the fact that Σ is umbilic. Thus this latter property is needed only to preserve convexity and pinching of the solutions M_{t}, cf. section 9. We begin with summarizing the assumptions we will need throughout this section:

Assumption 10.1 Let Σ be convex and umbilic and M_{0} a strictly convex, imbedded hypersurface. Furthermore, let F : [0, T_{C}) ⟶ IR^{n+1} be a solution to equation 1.2 on a maximal time interval and F (M^{n}, 0) = M_{0}.

10.1 Convergence to a Point

Here we generalize an idea of Tso [24] to the boundary case. Note that, by Theorem 4.5 and Proposition 9.4, the M_{t} will be imbedded and strictly convex with a uniform lower bound on the second fundamental form, i.e. there exists ε > 0 such that

h_{ij} ≥ ε g_{ij} ∀(x, t) ∈ M^{n} × [0, T_{C}) . (10.1)

Now for all t the hypersurfaces M_{t} and Σ bound a convex domain K_{t} ⊂ IR^{n+1}, and from equation 8.4 we immediately conclude

K_{t_{2}} ⊂ K_{t_{1}} if t_{2} ≥ t_{1} .

We set

K := ⋂ {K_{t} : 0 ≤ t < T_{C}}

and observe that K is convex as this holds for each K_{t}. Now suppose that σ > 0 and p ∈ K such that the ball B_{2 σ}^{n+1}(p) ⊂ IR^{n+1} is contained in K. Then, choosing coordinates in IR^{n+1} with origin in p, we obtain s ≥ 2 σ on M^{n} × [0, T_{C}) for the support function of our solution F. According to Theorem 8.8 this implies an upper bound on H and thus, by convextiy, on |A|. Hence, by the results of section 6, we can extend F to a solution on [0, T_{C} + δ) which contradicts the maximaltiy of T_{C}.

We therefore conclude that K cannot contain any ball B_{r}^{n+1}. But this implies that K has zero volume, V (K) = 0, as K is a bounded, convex domain. Now we can prove:

Theorem 10.2 (Convergence to a point) Under the assumptions 10.1 the hypersurfaces M_{t} converge to a point on Σ as t → T_{C}.

Proof: (cf. [24], Theorem 4.2) Suppose K is not a point, i.e. diam(K) > 0. Let p, q be such that d := dist_{n+1}(p, q) = diam(K).

We have p, q ∈ K_{t} ∀t, and as V (K) = 0 there is a (two dimensional) plane E through p, q such that the area of the convex subset C_{t} := K_{t} ∩ E ⊂ IR^{2} of E fulfils

(∗)  A(C_{t}) ⟶^{t→T_{C}} 0 .

As K_{t_{2}} ⊂ K_{t_{1}} whenever t_{2} ≥ t_{1}, we have

(∗∗)  C_{t_{2}} ⊂ C_{t_{1}} ∀ t_{2} ≥ t_{1} .

The convex curve γ_{t} := ∂C_{t} can be decomposed into γ_{t} = γt^{′} ∪ γt^{′′} such that γt^{′} ⊂ M_{t}, γt^{′′} ⊂ Σ (hereby γt^{′′} = ∅ being allowed) and (from (∗∗)):

(∗∗∗)  γt_{2}^{′′} ⊂ γt_{1}^{′′} ∀ t_{2} ≥ t_{1} .

This means that as t increases, no ”new” boundary points of Σ can occur on γ_{t}. As γ_{t} is convex, we conclude from (∗) that there must exist points on γ_{t} with smaller and smaller curvature as t increases. From (∗∗∗) it follows that these points must belong to γt^{′} ⊂ M_{t} which is a contradiction to the uniform convexity equation 10.1. Thus K must be a point, and the result follows. □

Proof of Proposition 1.5: In the case n = 1 strict convexity of the M_{t} is equivalent to H ≥ C_{0} > 0, and by Theorem 8.2 this condition is preserved for arbitrary convex hypersurfaces Σ. □

10.2 Convergence to a Hemisphere

Now we want to study the shape of the singularity; hereby, we follow an idea due to Hamilton [12] and rescale the entire flow. A similar rescaling for curves was carried out by Altschuler in [1]. In addition to Assumption 10.1, we will require n ≥ 2 in the sequel. Note that by Theorem 9.7 and Corollary 9.8, a pinching condition

h_{ij} ≥ ε H g_{ij}

will hold for t ∈ [0, T_{C}).

Definition 10.3 Let (x_{k}, t_{k}) be a sequence in M^{n} × [0, T_{C}) such that t_{k} ↗ T_{C}. If

H(x_{k}, t_{k}) = max{H(x, t_{k}) : x ∈ M^{n}} ∀k ,

(x_{k}, t_{k}) is called a blowup-sequence.

In the sequel we will rescale the mean curvature flow equation 1.2 along a given blowup-sequence. To this end, let

λ_{k} := H(x_{k}, t_{k}) = max_{M^{n}} H(., t_{k}) ,

α_{k} := −λ_{k}^{2} t_{k} ,
ω_{k} := λ_{k}^{2} (T_{C} − t_{k}) ,
τ := λ_{k}^{2} (t − t_{k}) ,
F̃_{k}(., τ) := λ_{k} (A_{k}·F (., t) + b_{k}) , A_{k} ∈ SO_{n+1}IR , b_{k} ∈ IR^{n+1} ,
M̃_{τ}^{(k)} := F̃_{k}(M^{n}, τ) .

Thereby let A_{k} and b_{k} be such that F̃_{k}(x_{k}, 0) = 0 and ν̃_{k}(x_{k}, 0) = e_{n+1} for any k. A straightforward calculation then yields that each F̃_{k} is itself a mean curvature flow, and

H̃_{k}(x_{x}, 0) = 1 ∀k .

As usual, a singularity will be referred to as of type I if max_{M^{n}} |A|^{2}(., t) ≤ K (T_{C} − t)^{−1} as t → T_{C} with a given constant K > 0; otherwise it will be called a type-II-singularity. Now, given a blowup-sequence, α_{k} ↘ −∞ as t → T_{C}. Moreover, by taking a subsequence, we can obtain ω_{k} ↗ ω < ∞ in the case of a type-I-singularity, and ω_{k} ↗ ∞, otherwise. Finally, as λ_{k} → ∞, we can assume λ_{k} > 1 ∀k.

Now the crucial step is to control the curvature of the rescaled hypersurfaces:

Proposition 10.4 (Curvature estimates for the rescaled hypersurfaces)

Type I: ∀δ > 0 ∃k_{0} = k_{0}(δ) and C = C(δ) such that
H̃_{k} ≤ C on M^{n} × [α_{k_{0}}, ω − δ] , ∀k ≥ k_{0} .
Type II: ∀ ω̄ > 0, ∀ε > 0 ∃k_{0} = k_{0}(ω̄, ε) such that
H̃_{k} ≤ 1 + ε on M^{n} × [α_{k_{0}}, ω̄] , ∀k ≥ k_{0} .

Hence on compact domains of spacetime, the curvature of the rescaled flows is bounded uniformly in k.

Proof:

I: As ω_{k} → ω, we can choose k_{0} such that [α_{k_{0}}, ω − δ] ⊂ [α_{k}, ω_{k} − \frac{δ}{2}] ∀k ≥ k_{0}. Now suppose the statement does not hold. Then there is a sequence (y_{k}, τ_{k}) ∈ M^{n} × [α_{k_{0}}, ω − δ] such that H̃_{k}(y_{k}, τ_{k}) → ∞. As for k ≥ k_{0}, ω_{k} ≥ ω − \frac{δ}{2} we can choose a subsequence with

H̃k̂(yk̂, τk̂) ≥ Ck̂ , yk̂ ∈ M^{n} , τk̂ ≤ ωk̂ − \frac{δ}{2} , Ck̂ ↗ ∞

and thereby, defining t̃k̂ := λ−2 ; k̂ τk̂ + tk̂ :

(∗)  H(yk̂, t̃k̂) ≥ λk̂ Ck̂ , yk̂ ∈ M^{n} , 0 ≤ t̃k̂ ≤ T^{C} − λ−2 ; k̂ \frac{δ}{2} .

But the singularity being of type I implies

H(yk̂, t̃k̂) ≤ \sqrt{\frac{2 K}{δ}} λk̂ ;

hence k̂ → ∞ yields a contradiction to (∗).

II: This has been proved by Hamilton [12] as follows: Given the sequence (x_{k}, t_{k}), choose (x̃_{k}, t̃_{k}) such that

(∗)  (t^{k} − t̃_{k}) H^{2}(x̃_{k}, t̃_{k}) = max_{M^{n}×[0,t_{k}]}(t^{k} − t) H^{2}(., t) .

Defining ω̃_{k} := λ_{k}^{2} (t_{k} −t̃_{k}), we obtain H̃_{k}(x̃_{k}, 0) = 1 ∀k and ω̃_{k} → +∞ as k → ∞. Furthermore, (∗) yields H̃_{k}^{2}(., τ) ≤ \frac{ω_{k}}{ω_{k}−τ} ∀ τ ∈ [α_{k}, ω_{k}) and thereby the result. □

Now the curvature bounds immediately imply local gradient bounds which are uniform in k: Let p_{0}^{(k)} := F̃_{k}(x_{0}, τ_{0}) , (x_{0}, τ_{0}) ∈ M^{n} × [α_{k}, ω_{k}] and Z_{r}(p_{0}^{(k)}) ⊂ IR^{n+1} denote the cylinder through p_{0}^{(k)} with axis ω_{k} := ν̃_{k}(x_{0}, τ_{0}) as in section 6. Furthermore, let

ṽ_{k}(x, τ) := ⟨ ω_{k} , ν̃_{k}(x, τ) ⟩^{−1} .

Then we can apply Theorems 6.7 and 6.13 to the rescaled flows F̃_{k} and obtain local gradient bounds on domains U_{r_{k}} × [τ_{0}, τ_{0} + Δ_{k}] with r_{k} and Δ_{k} depending only on the maximum of the curvatures |Ã_{k}| , |^{Σ}Ã_{k}| of the rescaled hypersurfaces M̃_{τ} , Σ̃_{k}. But as the M̃_{τ} are convex, the above Proposition yields a bound on |Ã_{k}| which is uniform in k. Moreover, as we assumed λ_{k} ≥ 1, the curvatures of the Σ̃_{k} will be uniformly bounded in k as well. Hence we obtain a local gradient estimate which is uniform in k. Now we can state the first Theorem concerning a limit flow:

Theorem 10.5 A subsequence of the rescaled flows F̃_{k} converges to a limit flow F̃_{∞} : M^{n} × (α, ω) ⟶ IR^{n+1} (ω = ∞ allowed) such that

(i) F̃ is a mean curvature flow;

(ii) If the M_{τ}^{∞} possess a boundary, then ∂M_{τ}^{∞} ⊂ Σ̃_{∞}, Σ̃_{∞} is a hyperplane and ⟨ν̃_{∞}, μ̃_{∞}⟩ = 0;

(iii) The hypersurfaces M̃_{τ}^{∞} are pinched as the original hypersurfaces, i.e. ε H g_{ij} ≤ h_{ij} ≤ κ H g_{ij} implies ε H̃_{∞} g̃_{ij}^{∞} ≤ h̃_{ij}^{∞} ≤ κ H̃_{∞} g̃_{ij}^{∞}.

Proof: In view of the above considerations, the only thing to check is the convergence of a subsequence. This can be done by standard methods involving the Arzelà-Ascoli Theorem: We first let A : B_{R}^{n+1}(0) × [τ_{−}, τ_{+}]. Using the above k-uniform local gradient estimate , we can cover A by finitely many spacetime-cylinders C_{r,Δ}(p_{m}, τ_{m}) such that for each (p_{m}, τ_{m}) and every k the intersection of the hypersurfaces M̃_{τ}^{k} with C_{r,Δ}(p_{m}, τ_{m}) can be written as a graph with bounded gradient. Hence the Arzelà-Ascoli Theorem yields the convergence of a subsequence on A with the desired properties. Doing this for a whole sequence A_{R^{l},[τ_{−}^{l},τ_{+}^{l}]} and choosing a diagonal sequence completes the proof. □

Remark 10.6 H := Σ̃_{∞} is a hyperplane if the hypersurfaces of the limit flow do have a boundary. Thus, in this case we can by reflection at H extend the hypersurfaces symmetrically to closed, convex and pinched hypersurfaces without boundary which again move under the mean curvature flow. Thus, in any case we can in the sequel assume the M̃_{τ}^{∞} to be complete, strictly convex, pinched hypersurfaces.

The limit flow can now be described by means of a result due to Hamilton ([11], Main Theorem):

Theorem 10.7 Let M be a smooth, strictly convex, complete hypersurface bounding a region in IR^{n+1} , n ≥ 2. Let M be ε-pinched, i.e. let h_{ij} ≥ ε H g_{ij}. Then M is compact.

We remark that this is the point where we need n ≥ 2. An immediate consequence of this Theorem is

Theorem 10.8 Under Assumption 10.1 the singularity at t = T_{C} is a type-I-singularity.

Proof: By Theorem 10.5, F̃_{∞} moves under the mean curvature flow. Thus the singularity being of Type II would imply that the solution would be defined for τ ∈ (−∞, ∞). But then all the M̃_{τ} would have to be noncompact, as compact hypersurfaces are contained in a ball B_{R} ⊂ IR^{n+1} and therefore cannot ”live” eternally. Now the M̃_{τ} are pinched by Theorem 10.5, and thus we run into a contradiction to Theorem 10.7. □

Hence the boundary of the M̃_{k} cannot be ”shot away” to infinity as k → ∞. But, up to now, it could shrink away such that the M̃_{k} would be compact, closed hypersurfaces. In order to rule out this possibility we observe that the image of the Gauss map of the ”original” hypersurfaces M_{t} is contained in a hemisphere. Hence this holds for each of the rescaled flows, the various hemispheres depending on k. Again by the Arzelà-Ascoli Theorem, we obtain the C^{∞}-convergence of a subsequence to a limit flow F̃_{∞} such that in addition to the above properties the images of all M̃_{τ}^{∞} are contained in a unique hemisphere. According to Theorem 10.5 the boundary of these hypersurfaces will therefore lie on a hyperplane. From there, the proof of Theorem 1.4 is an immediate consequence of Remark 10.6 and a Theorem of Huisken [13] which states that closed, strictly convex hypersurfaces moving by mean curvature shrink to a single point within finite time, the (appropriately) rescaled hypersurfaces thereby converging to a sphere in the C^{∞}-topology.

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