CONVEXITY ESTIMATES FOR FLOWS BY POWERS OF THE MEAN CURVATURE

FELIX SCHULZE

appendix with Oliver C. Schnürer

Abstract. We study the evolution of a closed, convex hypersurface in ℝ^{n+1} in direction of its normal vector, where the speed equals a power k ⩾ 1 of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvature at every point is close enough to 1, depending only on k and n, then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.

1. Introduction

In this paper we investigate the following problem. Let M^{n} be a smooth, compact manifold without boundary, and F_{0} : M^{n} → ℝ^{n+1} a smooth immersion which is convex. We study smooth families of immersions F (·, t) : M^{n} × [0, T) → ℝ^{n+1}, which satisfy

(⋆) \{ \begin{matrix}F (·, 0) = F_{0}(·) \\ \frac{dF}{dt} (·, t) = −H^{k}(·, t)ν(·, t) . \end{matrix}

where k > 0, H is the mean curvature and ν is the outer unit normal, such that −Hν = H⃗ is the mean curvature vector. Throughout the paper we will call such a flow an H^{k}-flow. In [9] we were able to show:

Theorem 1.1. Let F_{0} : M^{n} → ℝ^{n+1} be a smooth immersion with positive mean curvature, that is H(F_{0}(M^{n})) > 0. Then there exists a unique, smooth solution to the initial value problem (⋆) on a maximal, finite time interval [0, T). In the case that

i) F_{0}(M^{n}) is strictly convex for 0 < k < 1 ,
ii) F_{0}(M^{n}) is weakly convex for k ⩾ 1 ,

then the surfaces F (M^{n}, t) are strictly convex for all t > 0 and they contract for t → T to a point in ℝ^{n+1}.

Here ’weakly convex’ and ’strictly convex’, resp. are defined as all the eigenvalues of the second fundamental form being nonnegative, and positive, resp. In this paper we want to present the following extension of the previous result. Let K denote the Gauss curvature. To control the pinching of the initial hypersurface we use that by the arithmetic-geometric mean inequality we have

2000 Mathematics Subject Classification. 53C44, 35B40.

The author is a member of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.

0 ⩽ K/H^{n} ⩽ 1/n^{n},

with equality on the right side if and only if all eigenvalues are equal.

Theorem 1.2. For k ⩾ 1 there exists a nonnegative constant C(n, k) < 1/n^{n} such that the following holds: If the initial hypersurface is pinched in the sense that

\frac{K(p)}{H^{n}(p)} > C(n, k) for all p ∈ M,

then this is preserved under the H^{k}-flow. The constant C(n, k) is increasing in k, lim_{k↘1} C(n, k) = 0 and lim_{k→∞} C(n, k) = 1/n^{n}. Furthermore the rescaled embeddings

F̃(τ, p) := ((k + 1)n^{k}(T − t))^{−1/(k+1)} (F (τ, p) − x_{0})

converge for τ → ∞ exponentially in the C^{∞}-topology to the unit sphere. Here τ := −(k + 1)^{−1}n^{−k} log(1 − t/T), where T is the maximal time of existence of the unrescaled flow and x_{0} is the point in ℝ^{n+1} where the surfaces contract to.

Together with O. Schnürer we give in the appendix a further extension in the case of 2-dimensional hypersurfaces in ℝ^{3}. For 1 ⩽ k ⩽ 5 we show that no initial pinching condition is needed to ensure that the rescaled embeddings as described above converge to the unit sphere. For k = 1 the flow considered in (⋆) is the mean curvature flow. In that case the statements of both Theorems are implied by the results of G. Huisken in [7], who showed that convex surfaces remain convex under the flow and contract to a ’round’ point in finite time. Similar results were obtained by B. Chow for the nth root of the Gauss curvature in [4] and for the square root of the scalar curvature in [5]. B. Andrews extended these results in [1] to a whole class of normal velocities which are homogeneous of degree one in the principal curvatures. For normal velocities which have a degree of homogeneity greater than one B. Chow proved a result similar to Theorem 1.2 for powers of the Gauss curvature in [4]. The behavior of the initial pinching condition in Chow’s result in the degree of homogeneity is analogous to our result here. Evolving 2-dimensional hypersurfaces more is known. In [2], B. Andrews shows that convex surfaces moving by Gauss curvature converge to round points without any initial pinching condition. O. Schnürer in [8] obtained similar results for a list of different normal velocities with homogeneity greater than one, including the cases H^{2}, H^{3}, H^{4}. For general speeds of higher homogeneity, B. Andrews showed in [3] that an initial pinching condition, depending only on the degree of homogeneity, ensures the convergence to a round point.

The rest of the paper is organized as follows. In section 2 we compute the evolution equation of the quantity K/H^{n}, and applying the maximum principle we can deduce that if the surface is initially pinched good enough it remains so under the flow. Further investigation shows that that the pinching improves if the mean curvature explodes. Interestingly this follows again by an application of the maximum principle and doesn’t need integral estimates as in the case of the mean curvature flow.

In section 3 we define a natural rescaling of the flow to show that the surfaces become more and more spherical as they contract to a point. Since the rescaled flow might not be anymore uniformly parabolic, we cannot directly apply estimates of Krylov-Safanov type to deduce smooth convergence to a sphere. But since the evolution equation for the mean curvature can be written in the form of a porous medium equation, we can circumvent this using Hölder estimates for this type of equations. In the appendix we present the result in the 2-dimensional case.

2. Pinching Estimates

We first state the evolution equations for geometric quantities like the induced metric g_{ij}, the induced measure dμ, the second fundamental form h_{ij} or equivalently the Weingarten map W_{p} = {h_{j}^{i}} : T_{p}M → T_{p}M. Further important quantities are the mean curvature H = g^{ij}h_{ij}, the norm of the of the second fundamental form squared |A|^{2} = h^{ij}h_{ij} and the Gauss curvature K = det(h_{j}^{i}). In the following we will always assume that our evolving surfaces are strictly convex, such that H, K > 0 and the inverse of the Weingarten map (b_{j}^{i}) := (h_{j}^{i})^{−1} is well-defined. In the case k ⩾ 1, Theorem 1.1 yields that the evolving surfaces are always strictly convex for positive times.

Lemma 2.1. The following evolution equations hold.

(i) \frac{∂}{∂t} g_{ij} = −2H^{k}h_{ij} ,
(ii) \frac{∂}{∂t} dμ = −H^{k+1} dμ ,
(iii) \frac{∂}{∂t} h_{j}^{i} = kH^{k−1}Δh_{j}^{i} + k(k − 1)H^{k−2}∇^{i}H∇_{j}H − (k − 1)H^{k}h_{l}^{i}h_{j}^{l}

+ kH^{k−1}|A|^{2}h_{j}^{i} ,

(iv) \frac{∂}{∂t} H^{l} = kH^{k−1}ΔH^{l} + lk(k − l)H^{k+l−3}|∇H|^{2} + l|A|^{2}H^{l+k−1}, l ∈ ℝ ,
(v) \frac{∂}{∂t} |A|^{2} = kH^{k−1}Δ|A|^{2} − 2kH^{k−1}|∇A|^{2} + 2k(k − 1)H^{k−2}h_{i}^{j}∇^{i}H∇_{j}H
− 2(k − 1)H^{k}tr (A^{3}) + 2kH^{k−1}|A|^{4} ,
(vi) \frac{∂}{∂t} K = kH^{k−1}ΔK − kH^{k−1}K^{−1}|∇K|^{2} − kH^{k−1}K∇^{i}b^{lm}∇_{i}h_{lm}

+ k(k − 1)H^{k−2}Kb^{lm}∇_{l}H∇_{m}H − (k − 1)H^{k+1}K

+ knH^{k−1}|A|^{2}K

= kH^{k−1}ΔK − \frac{k(n − 1)}{n}H^{k−1} \frac{|∇K|^{2}}{K} − \frac{k}{n}H^{2n+k−1} \frac{|∇(KH^{−n})|^{2}}{K}

+ k(k − 1)H^{k−2}Kb^{lm}∇_{l}H∇_{m}H + kH^{k−3}K|H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} − (k − 1)H^{k+1}K + knH^{k−1}|A|^{2}K ,

where

|H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} := g^{ij}b^{km}b^{ln}(H∇_{i}h_{mn} − h_{mn}∇_{i}H)(H∇_{j}h_{kl} − h_{kl}∇_{j}H) .

Proof. (i)−(v) follows from a direct calculation as for example in [1]. The first two lines in (vi) are again a direct calculation. The second equality then follows from the identity

\frac{1}{nK} |∇K|^{2} + K∇^{i}b^{lm}∇_{i}h_{lm} = −\frac{K}{H^{2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} + \frac{H^{2n}}{nK} |∇(KH^{−n})|^{2} .

□

An appropriate test function to control the pinching of the principal curvatures along the flow turns out to be K/H^{n}.

Lemma 2.2. The quantity K/H^{n} satisfies the evolution equation

\frac{∂}{∂t}(\frac{K}{H^{n}} ) = kH^{k−1}(Δ(\frac{K}{H^{n}} ) + \frac{(n + 1)}{nH^{n}}∇_{i}(\frac{K}{H^{n}} )∇_{i}H^{n} − \frac{(n − 1)}{nK}∇_{i}(\frac{K}{H^{n}} )∇_{i}K

− \frac{H^{n}}{nK} |∇(\frac{K}{H^{n}} )|^{2} + \frac{K}{H^{n+2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2}

+ (k − 1)\frac{K}{H^{n+1}} (b^{ij} − \frac{n}{H} g^{ij})∇_{i}H∇_{j}H + \frac{(k − 1)}{k} \frac{K}{H^{n}} (n|A|^{2} − H^{2})) .

Proof. This follows from computing

\frac{∂}{∂t}(\frac{K}{H^{n}} ) = \frac{1}{H^{n}} \frac{∂}{∂t}K − \frac{K}{H^{2n}} \frac{∂}{∂t}H^{n}

and the evolution equations of K and H^{n} above. Note that we can write

\frac{1}{H^{n}} ΔK − \frac{K}{H^{2n}} ΔH^{n} − \frac{(n − 1)}{n} \frac{1}{KH^{n}} |∇K|^{2} =

Δ(\frac{K}{H^{n}} ) + \frac{(n + 1)}{nH^{n}}∇_{i}(\frac{K}{H^{n}} )∇_{i}H^{n} − \frac{(n − 1)}{nK}∇_{i}(\frac{K}{H^{n}} )∇_{i}K − n(n − 1)\frac{K}{H^{n+2}} |∇H|^{2} .

□

We now aim to apply the maximum principle and show that that min_{p∈M} \frac{K}{H^{n}} (p, t) is non-decreasing in t. We argue as in [4]. Since always |A|^{2} ⩾ \frac{1}{n}H^{2}, the lowest order terms have the right sign. From [7] we know that if h_{ij} ⩾ εHg_{ij}, for some ε > 0, then

|H∇_{i}h_{kl} − ∇_{i}Hh_{kl}|_{g}^{2} ⩾ ε^{2} \frac{n − 1}{2}H^{2}|∇H|^{2} .

So by estimating b^{ij} ⩾ g^{ij}/H we obtain

\frac{K}{H^{n+2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} ⩾ ε^{2} \frac{n − 1}{2} \frac{K}{H^{n+2}} |∇H|^{2} .

Let C(n, k) denote the minimal constant such that 0 ⩽ C(n, k) < 1/n^{n} and for an ε ⩾ 0 it holds

\frac{K}{H^{n}} ⩾ C(n, k) ⟹ h_{ij} ⩾ εHg_{ij} and |b^{ij} − \frac{n}{H} g^{ij}| ⩽ \frac{ε^{2}}{2(k − 1)H} .

It then follows that

(k − 1)\frac{K}{H^{n+1}} |(b^{ij} − \frac{n}{H} g^{ij})∇_{i}H∇_{j}H| ⩽ ε^{2} \frac{n − 1}{2} \frac{K}{H^{n+2}} |∇H|^{2} ,

which yields:

Corollary 2.3. For k ⩾ 1 there exists a constant 0 ⩽ C(n, k) < 1/n^{n} such that the following holds: If the initial hypersurface is pinched in the sense that

\frac{K(p)}{H^{n}(p)} ⩾ C(n, k) for all p ∈ M,

then this is preserved under the H^{k}-flow. The constant C(n, k) is increasing in k, lim_{k↘1} C(n, k) = 0 and lim_{k→∞} C(n, k) = 1/n^{n}.

A next step is to show that at points where the mean curvature is big, the principal curvatures approach each other. To do this we define

f := \frac{1}{n^{n}} − \frac{K}{H^{n}} and f_{σ} := H^{σ}f .

Note that 0 ⩽ f ⩽ 1/n^{n} and f(p, t) = 0 iff the principal curvatures at (p, t) are all equal.

Lemma 2.4. The quantity f_{σ} has the evolution equation

\frac{∂f_{σ}}{∂t} = kH^{k−1}(Δf_{σ} + 2(1 − σ(1 + \frac{H^{n}}{K} f))⟨∇f_{σ}, \frac{∇H}{H} ⟩ + \frac{H^{n−σ}}{K} |∇f_{σ}|^{2}

+ H^{σ}(σ(k − 2 + σ(1 + f \frac{H^{n}}{K} ))f \frac{|∇H|^{2}}{H^{2}} − \frac{K}{H^{n+2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2}

− \frac{(k − 1)K}{H^{n+1}} (b^{ij} + \frac{n}{H} g^{ij})∇_{i}H∇_{j}H − \frac{(k − 1)K}{kH^{n}} (n|A|^{2} − H^{2}) + \frac{σ}{k} f|A|^{2})) .

Proof. From the preceding Lemma and Lemma 2.1 we have

\begin{aligned}\frac{∂f_{σ}}{∂t} = & H^{σ} \frac{∂f}{∂t} + σH^{σ−1}f \frac{∂H}{∂t} \\ = & kH^{k−1}(H^{σ}Δf + σH^{σ−1}fΔH + H^{σ}(\frac{n − 1}{H}⟨∇f, ∇H⟩ \\ & − \frac{n − 1}{nK}⟨∇f, ∇K⟩ + \frac{K}{H^{n+2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} + \frac{H^{n}}{nK} |∇f|^{2} \\ & + (k − 1)σf \frac{|∇H|^{2}}{H^{2}} − \frac{(k − 1)K}{H^{n+1}} (b^{ij} − \frac{n}{H} g^{ij})∇_{i}H∇_{j}H \\ & − \frac{(k − 1)K}{kH^{n}} (n|A|^{2} − H^{2}) + \frac{σ}{k} f|A|^{2})) . \end{aligned} ; (1)

We then compute

∇_{i}f_{σ} = H^{σ}∇_{i}f + σ \frac{f_{σ}}{H} ∇_{i}H
∇_{i}K = −H^{n−σ}∇_{i}f_{σ} + (n\frac{K}{H} + σH^{n−1}f)∇_{i}H
H^{σ}⟨∇f, ∇H⟩ = ⟨∇f_{σ}, ∇H⟩ − σ \frac{f_{σ}}{H} |∇H|^{2}
⟨∇K, ∇f⟩ = −H^{n−2σ}|∇f_{σ}|^{2} + H^{−σ}(n\frac{K}{H} + 2σH^{n−1}f)⟨∇H, ∇f_{σ}⟩

− σ(nK + σH^{n}f)f \frac{|∇H|^{2}}{H^{2}}

|∇f|^{2} = H^{−2σ}(|∇f_{σ}|^{2} − 2σ \frac{f_{σ}}{H} ⟨∇f_{σ}, ∇H⟩ + σ^{2}f_{σ}^{2} \frac{|∇H|^{2}}{H^{2}} )

Δf_{σ} = H^{σ}Δf + 2σH^{σ−1}⟨∇f, ∇H⟩ + σ(σ − 1)f \frac{|∇H|^{2}}{H^{2−σ}} + σH^{σ−1}fΔH

= H^{σ}Δf + σH^{σ−1}fΔH + \frac{2σ}{H} ⟨∇f_{σ}, ∇H⟩ − σ(σ + 1)f_{σ} \frac{|∇H|^{2}}{H^{2}} .

Inserting these computations in (1) and collecting terms gives the stated evolution equation. □

The next lemma will be needed to show that for small σ the lowest order terms have the right sign.

Lemma 2.5. Assume that λ_{i} ⩾ εH > 0 for some ε > 0 and for all i = 1 … n . Then there exists a δ > 0 such that

\frac{n|A|^{2} − H^{2}}{H^{2}} ⩾ δ(\frac{1}{n^{n}} − \frac{K}{H^{n}} ) .

Proof. Normalize the eigenvalues by defining

λ̃_{i} := \frac{λ_{i}}{H} .

Then ε ⩽ λ̃_{i} ⩽ 1, i = 1 … n and ∑_{i=1}^{n} λ̃_{i} = 1. Since n|A|^{2} − H^{2} = ∑_{i<j}(λ_{i} − λ_{j})^{2} we can write the desired inequality as

∑_{i<j}(λ̃_{i} − λ̃_{j})^{2} ⩾ δ(\frac{1}{n^{n}} − K(λ̃)) .

Both sides are positive and they are zero if and only if (λ̃_{1}, …, λ̃_{n}) = (1/n, …, 1/n). We do a Taylor expansion of both sides around (1/n, …, 1/n) on the hyperplane {∑_{i=1}^{n} λ̃_{i} = 1}. Since the left hand side is a polynomial of second order, its hessian has to be strictly positive and the claim follows by the compactness of the set {∑_{i=1}^{n} λ̃_{i} = 1, λ̃_{j} ⩾ ε, j = 1, …, n}. □

Applying the maximum principle leads to the following theorem.

Theorem 2.6. If k > 1 and the initial hypersurface is pinched in the sense that

(2)  \frac{K(p)}{H^{n}(p)} > C(n, k) ∀ p ∈ M,

then there exists a σ > 0 such that

(3)  f_{σ}(p, t) ⩽ max_{p∈M} f_{σ}(p, 0) ∀ (p, t) ∈ M × [0, T) .

Proof. To apply the maximum principle to the evolution equation of f_{σ}, we have to show that

\begin{matrix}σ(k − 2 + σ(1 + f \frac{H^{n}}{K} ))f \frac{|∇H|^{2}}{H^{2}} − \frac{K}{H^{n+2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} \\ −\frac{(k − 1)K}{H^{n+1}} (b^{ij} + \frac{n}{H} g^{ij})∇_{i}H∇_{j}H − \frac{(k − 1)K}{kH^{n}} (n|A|^{2} − H^{2}) + \frac{σ}{k} f|A|^{2} & ⩽ 0 . \end{matrix} ; (4)

By Lemma 2.3 the lower bound

(5)  \frac{K(p)}{H^{n}(p)} ⩾ C_{0}

is preserved, so as in the proof of Lemma there exists a η > 0 such that

−\frac{K}{H^{n+2}} |H∇_{i}h_{mn} − h_{mn}∇_{i}H|_{g,b}^{2} − \frac{(k − 1)K}{H^{n+1}} (b^{ij} + \frac{n}{H} g^{ij})∇_{i}H∇_{j}H ⩽ −η \frac{|∇H|^{2}}{H^{2}} .

Choosing σ small enough the terms involving ∇H in (4) have the right sign. Since |A^{2}| ⩽ H^{2} and (5) implies that λ_{i} ⩾ εH for some ε > 0, we conclude by Lemma 2.5 that also

−\frac{(k − 1)K}{kH^{n}} (n|A|^{2} − H^{2}) + \frac{σ}{k} f|A|^{2} ⩽ 0

for σ maybe chosen even smaller. □

3. Rescaling and Convergence

In this chapter we consider a natural rescaling of the evolution equation (⋆). It agrees with the rescaling for the mean curvature flow of convex surfaces in [7] and for the general class of flows treated in [1]. To be able to apply Theorem 2.6 we will henceforth assume that k > 1 and that the initial surface is pinched in the sense of (2). For k = 1, i.e. the mean curvature flow, the results are known by the work of Huisken [7]. The next section follows closely Chapter 7 of [1].

For the evolution of a sphere with initial radius R_{0} we can compute that R(t) = ((k + 1)n^{k}(T − t))^{1/(k+1)}, where T = R_{0}^{k+1}/(n^{k}(k +1)) is the maximal time of existence. This suggests the following general rescaling. Assume we have a general convex solution of (⋆) on a maximal time interval [0, T). By Theorem 1.1 we know that the surfaces contract to a point x_{0} in ℝ^{n+1} as t → T. Define α := (k + 1)n^{k} , and a new time parameter τ by τ := −\frac{1}{α} log(1 − \frac{t}{T}). The rescaled immersions F̃_{τ} := (α(T − t))^{−1/(k+1)}(F_{t} − x_{0}) then satisfy the evolution equation

(6)  \frac{∂F̃}{∂τ} (p, τ) = −H̃^{k}(p, τ)ν̃(p, τ) + n^{k}F̃(p, τ) for τ ∈ [0, ∞) .

In following we will distinguish geometric quantities associated with the rescaled immersions by a tilde.

Lemma 3.1. The evolution equations for the rescaled metric g̃_{ij}, the induced measure dμ̃ and the rescaled mean curvature H̃ are

(i)  \frac{∂}{∂τ} g̃_{ij} = −2H̃^{k}h̃_{ij} + 2n^{k}g̃_{ij}
(ii)  \frac{∂}{∂τ} dμ̃ = −H̃^{k+1}dμ̃ + n^{k+1}dμ̃
(iii)  \frac{∂}{∂τ} H̃ = ΔH̃^{k} + |Ã|^{2}H̃^{k} − n^{k}H̃ .

To control the convergence we relate the principal curvatures to the inner radius ρ_{−} and the outer radius ρ_{+}, which are defined as

ρ_{+}(t) := inf{r : B_{r}(y) encloses F_{t}(M) for some y ∈ ℝ^{n+1}}
ρ_{−}(t) := sup{r : B_{r}(y) is enclosed by F_{t}(M) for some y ∈ ℝ^{n+1}}

where B_{r}(y) is the ball with radius r centered at y. By Corollary 2.3 we know that there is a constant C_{1} such that

λ_{max}(p, t) ⩽ C_{1}λ_{min}(p, t) ∀ (p, t) ∈ M × [0, T) .

Lemma 5.4 in [1] then guarantees a constant C_{2} such that

ρ_{+} ⩽ C_{2}ρ_{−} .

Lemma 3.2.

\frac{1}{C_{2}} ⩽ ρ̃_{−} ⩽ 1 ⩽ ρ̃_{+} ⩽ C_{2} ,

for all τ ⩾ 0.

The proof of this Lemma is nearly identical to the proof of Lemma 7.2 in [1], just the rescaling is different.

Now Lemma 4.5 in [9] gives that

H(p, t) ⩽ \frac{3}{2} \frac{n(k + 1)}{k} \frac{1}{δ} for (p, t) ∈ M × [0, T′]

if all surfaces M_{t}, t ∈ [0, T′] enclose a fixed Ball B_{δ}(y_{0}) for some y_{0} ∈ ℝ^{n+1}, δ small enough. This gives

sup_{(p,t)∈M×[0,T′]} H(p, t) ⩽ \frac{3}{2} \frac{n(k + 1)}{k} \frac{1}{ρ_{−}(T′)}

and we obtain

(7)  H̃(τ) ⩽ C_{3} ∀ τ ∈ [0, ∞) ,

i.e. we have a uniform bound on |Ã|^{2}. To obtain C^{2,α}-estimates we cannot apply the results of Krylov, since we don’t know of a suitable Harnack-inequality for this flow which guarantees a positive lower bound for H̃ and so ensures uniform parabolicity. We circumvent this by noting that the evolution equation for H̃ can be written in the form of a porous medium equation, see Lemma 3.1 (iii). In [6] E. DiBenedetto and A. Friedman have established interior Hölder-estimates for such equations, which we aim to apply.

Lemma 3.3. For a rescaled flow, there is a constant C_{4} = C(n, k, C_{3}) such that

∫_{τ_{1}}^{τ_{2}} |∇H̃^{k}|^{2} dμ̃ dτ ⩽ C_{4}(1 + τ_{2} − τ_{1})

for 0 ⩽ τ_{1} < τ_{2} < ∞.

Proof. Computing

\frac{∂}{∂τ} ∫ H̃^{k+1} dμ̃

together with integration by parts leads to the inequality

\begin{matrix}∫ |∇H̃^{k}|^{2} dμ̃ ⩽ \frac{−1}{k + 1} \frac{∂}{∂τ} ∫ H̃^{k+1} dμ̃ + & \frac{k}{k + 1} ∫ H̃^{2k+2} dμ̃ \\ + & n^{k} \frac{n − k − 1}{k + 1} ∫ H̃^{k+1} dμ̃ . \end{matrix}

Then integrate this inequality. By equation (7) the mean curvature H̃ is uniformly bounded, and since M̃_{τ} is convex it holds that

μ̃(M̃_{τ}) ⩽ nω_{n+1}ρ̃_{+}^{n} .
□

Lemma 3.4. There are constants C_{5}, η > 0 and 0 < α < 1 such that for every (p, τ) ∈ M × (η, ∞) the α-Hölder norm in space-time of H̃ on B_{η}(p) × (τ − η, τ + η) is bounded by C_{5}.

Proof. Pick a (p_{0}, τ_{0}) ∈ M × [0, ∞). By Lemma 3.2 there is a y ∈ ℝ^{n+1} such that B_{1/C_{2}}(y) is enclosed by M̃_{τ_{0}}. Since the speed of the rescaled evolution is uniformly bounded, there is a ζ > 0, not depending on (p_{0}, τ_{0}), such that M̃_{τ} encloses B_{1/(2C_{2})}(y) for all τ ∈ [τ_{0} − ζ, τ_{0} + ζ]. Since all the M̃_{τ} are convex, we have that

⟨ν̃, F̃ − y⟩(p, τ) ⩾ \frac{1}{2C_{2}} ∀ (p, τ) ∈ M × [τ_{0} − ζ, τ_{0} + ζ] .

Because also ρ̃_{+} is uniformly bounded, there is a uniform constant η > 0, 2η ⩽ ζ, such that the evolving surfaces M̃_{τ} ∩ ((B_{2η}(p_{0}) ∩ T_{p_{0}} M̃_{τ_{0}}) × ℝ · ν̃(p_{0}, τ_{0})) can be written as a graph of a function u(x, τ) on B_{2η}(p_{0}) ∩ T_{p_{0}} M̃_{τ_{0}}, which is uniformly bounded in C^{2}. The evolution equation of H̃ in this coordinates can then be written in the form

\frac{∂}{∂τ} H̃(x, τ) = D_{i}((δ^{ij} − \frac{D^{i}uD^{j}u}{1 + |Du|^{2}} )D_{j}H̃^{k}) + b^{i}D_{i}H̃^{k} + c ,

where b^{i} = b^{i}(x, u, Du, D^{2}u) and c = c(x, u, Du, D^{2}u), which are uniformly bounded. From Lemma 3.3 we have

∫_{τ_{0}−2η}^{τ_{0}+2η} ∫_{B_{2η}(p_{0})∩T_{p_{0}} M̃_{τ_{0}}} |DH̃^{k}|^{2} dx ⩽ C .

These are all requirements to apply Theorem 1.2 in [6], which gives the claimed interior Hölder estimates. □

We now take a sequence of times (τ_{j}) with τ_{j} → ∞. Since the second fundamental form is uniformly bounded we can extract a subsequence of times, again denoted by (τ_{j}), such that

M̃_{τ_{j}} → M_{∞} ,

in C^{1,α} and M_{∞} is a convex C^{1,1}-surface. Because ρ̃_{+} ⩽ C_{2} there exist points p_{j}, such that

H̃(p_{j}, τ_{j}) ⩾ \frac{n}{C_{2}} .

Assume F̃(p_{j}, τ_{j}) → p_{∞} ∈ M_{∞}. By Lemma 3.4 there is a δ > 0 such that

H̃|_{B_{δ}(p_{j})∩M̃_{τ_{j}}} ⩾ \frac{n}{2C_{2}} .

Using parabolic Schauder estimates we obtain locally uniform C^{∞}-estimates as well as the convergence of

M̃_{τ_{j}} ∩ B_{δ}(p_{∞}) → M_{∞} ∩ B_{δ}(p_{∞})

in C^{∞}. By Theorem 2.6 it follows that M_{∞} ∩ B_{δ}(p_{∞}) is totally umbilic, and by this part of a sphere with mean curvature grater or equal to n/C_{2}. Repeating this argument with a limit point q_{∞} ∈ B_{δ}(p_{∞}) \ B_{δ/2}(p_{∞}) we can extend the region where M_{∞} is known to be spherical, and after finitely many steps we obtain that M_{∞} is a sphere. As in Corollary 7.15 in [1] we obtain that M_{∞} is a sphere with radius 1, centered at the origin. So the limit surface is independent of the approximating sequence and the whole flow converges in C^{∞}.

Theorem 3.5. The immersion F̃_{τ} converges exponentially to a limit immersion F̃_{∞} with image equal to the unit sphere. Where under exponential convergence it is understood that there exist constants δ_{i}, C_{i} > 0, i ⩾ 0 such that

| log (\frac{g̃_{∞}(ξ, ξ)}{g̃_{τ} (ξ, ξ)})| ⩽ C_{0}e^{−δ_{0}τ} for every non-zero tangent vector ξ,
|∇^{(i)}(Ã_{τ} − Ã_{∞})| ⩽ C_{i}e^{−δ_{i}τ} ,
|H̃_{τ} − n| ⩽ C_{0}e^{−δ_{0}τ} .

Proof. Let again f̃:= 1/n^{n} − K̃/H̃^{n}. By Lemma 2.4 and Lemma 2.5 we have that

\frac{∂}{∂τ} f̃(τ) ⩽ kH̃^{k−1}(Δf̃+ \frac{2}{H̃} ⟨∇f̃, ∇H̃⟩ + \frac{H̃^{n}}{K̃} |∇f̃|^{2} − δH̃^{2}f̃)

for δ > 0 small enough. Since H̃^{2} ⩾ 1/(4n^{2}) for τ big enough, there exists a δ′ > 0 and a constant C, such that

f̃(τ) ⩽ Ce^{−δ′τ} .

With a similar argument as in Lemma 2.5 we can deduce that this implies

|λ̃_{max} − λ̃_{min}|(p, τ) ⩽ Ce^{−δ′τ} ∀ p ∈ M.

Now write w_{ij} := h̃_{ij} − \frac{H̃}{n} g̃_{ij} and observe that

|w|^{2} = w^{ij}w_{ij} = |Ã|^{2} − \frac{1}{n}H̃^{2} = \frac{1}{n} ∑_{i≠j}(λ̃_{i} − λ̃_{j})^{2} ⩽ Ce^{−2δ′τ} .

This gives by interpolation that

|∇_{k}w_{ij}| ⩽ Ce^{−δ′′τ} i, j, k = 1, …, n ,

for some 0 < δ′′ ⩽ δ′. Assume we have g̃_{ij} = δ_{ij} at a point p, then

∇_{k}H̃ = ∑_{i} ∇_{k}h̃_{ii} = ∑_{i} ∇_{k}h̃_{ki} = \frac{∇_{k}H̃}{n} + ∑_{i} ∇_{i}w_{ki} ,

which gives

|∇_{k}H̃| ⩽ \frac{n}{n − 1} ∑_{i} |∇_{i}w_{ki}| ⩽ Ce^{−δ′′τ} .

Similarly one obtains

|∇Ã| ⩽ Ce^{−δ′′′τ} ,

and all higher derivatives by interpolation. The remaining estimates on curvature follow by writing the converging surfaces M̃_{τ} as normal graphs over the unit sphere and using interpolation again. The estimate on the metric and the convergence of the immersions is again as in [1]. □

Appendix A. The 2-dimensional Case

with Oliver C. Schnürer

In this appendix we want to show that in the case of 2-dimensional surfaces in ℝ^{3}, for 1 ⩽ k ⩽ 5, one can drop the initial pinching condition in Theorem 1.2. In concise form the theorem is stated as follows.

Theorem A.1. A smooth closed convex surface in ℝ^{3}, contracting with normal velocity H^{k}, where 1 ⩽ k ⩽ 5, contracts to a round point in finite time.

The case k = 1 is again the well-known mean curvature flow, see [7], and the cases k = 2, 3, 4 were already treated in [8]. The proof of this Theorem depends crucially on the following pinching estimate.

Lemma A.2. For a family of smooth closed strictly convex surfaces M_{t} in ℝ^{3}, flowing with normal velocity H^{k}, where 1 ⩽ k ⩽ 5, the quantity

(8)  max_{M_{t}} \frac{(λ_{1} + λ_{2})^{2k}(λ_{1} − λ_{2})^{2}}{4λ_{1}^{2}λ_{2}^{2}}

is non-increasing in time.

Here λ_{1}, λ_{2} denote the principal curvatures of M_{t}. This monotone quantity was found by O. Schnürer using a sieve algorithm implemented into a computer program. The algorithm uses random numbers for the test, whether the “right-hand side” of the evolution equation of such a quantity is non-positive and thus the maximum principle can be applied to prove monotonicity. For a detailed discussion of this algorithm see [8], where this method is also applied to find monotone quantities for a whole list of different normal velocities. The proof of Lemma A.2 follows closely the proofs in the paper cited above.

Proof of Lemma A.2. Set

w := \frac{H^{2k}(2|A|^{2} − H^{2})}{(H^{2} − |A|^{2})^{2}} = \frac{(λ_{1} + λ_{2})^{2k}(λ_{1} − λ_{2})^{2}}{4λ_{1}^{2}λ_{2}^{2}} .

For the rest of the proof we consider a point, where w|_{M_{t}} attains a positive maximum for some t > 0. By the maximum principle, it suffices to show that w̃ := log w is non-increasing in such a point for our Theorem to follow. Now rewrite

w̃ = 2 log(H^{k}) + log(2|A|^{2} − H^{2}) − 2 log(H^{2} − |A|^{2})
≡ 2 log A + log B − 2 log C .

In a critical point of w̃, we obtain

(\frac{d}{dt} − kH^{k−1}Δ)w̃ = \frac{2}{A}(\frac{d}{dt} − kH^{k−1}Δ)A + \frac{1}{B} (\frac{d}{dt} − kH^{k−1}Δ)B

− \frac{2}{C} (\frac{d}{dt} − kH^{k−1}Δ)C + \frac{k}{2} H^{k−1} \frac{1}{B^{2}} |∇B|^{2}

− \frac{2k}{AB} H^{k−1}⟨∇A, ∇B⟩

and for i = 1, 2

2AB ∇_{i}C = 2CB ∇_{i}A + AC ∇_{i}B ,

which yields

(9)  ∇_{i}h_{22} = \frac{λ_{2}^{2}H + kλ_{1}λ_{2}(λ_{1} − λ_{2})}{λ_{1}^{2}H − kλ_{1}λ_{2}(λ_{1} − λ_{2})}∇_{i}h_{11} =: \frac{α}{β}∇_{i}h_{11} ,

where we assume for the moment that α, β ≠ 0. Here we have chosen normal coordinates such that the second fundamental form is diagonal, i.e. h_{11} = λ_{1}, h_{22} = λ_{2}, h_{12} = h_{21} = 0. We now insert the evolution equations from Lemma 2.1 and combine the above results to obtain

(\frac{d}{dt} − kH^{k−1}Δ)w̃ = \frac{2}{A}(\frac{d}{dt} − kH^{k−1}Δ)H^{k} + (\frac{2}{B} + \frac{2}{C} )(\frac{d}{dt} − kH^{k−1}Δ)|A|^{2}

− (\frac{1}{B} + \frac{2}{C} )(\frac{d}{dt} − kH^{k−1}Δ)H^{2} + \frac{k}{2} \frac{H^{k−1}}{B^{2}} |∇(h_{11} − h_{22})^{2}|^{2}

− 2k \frac{H^{k−1}}{AB} ⟨∇H^{k}, ∇(h_{11} − h_{22})^{2}⟩

= 2kH^{k−1}|A|^{2} + \frac{2|A|^{2}}{(2|A|^{2} − H^{2})(H^{2} − |A|^{2})}( − 2kH^{k−1}|∇A|^{2}

+ 2k(k − 1)H^{k−2}h^{ij}∇_{i}H∇_{j}H − 2(k − 1)H^{k}tr(A^{3})

+ 2kH^{k−1}|A|^{4}) − \frac{3|A|^{2} − H^{2}}{(2|A|^{2} − H^{2})(H^{2} − |A|^{2})}(2H^{k+1}|A|^{2}

+ 2k(k − 2)H^{k−1}|∇H|^{2}) + 2k \frac{H^{k−1}}{(λ_{1} − λ_{2})^{2}} |∇(h_{11} − h_{22})|^{2}

− \frac{4k^{2}}{(λ_{1} − λ_{2})}H^{k−2}⟨∇H, ∇(h_{11} − h_{22})⟩ .

The terms not containing derivatives of h_{ij} cancel. We multiply the RHS by (2|A|^{2}− H^{2})(H^{2} − |A|^{2})H^{2−k} and use (9):

RHS · \frac{2(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}}{H^{k−2}} = − 4kH|A|^{2}|∇A|^{2} + 4k(k − 1)|A|^{2}h^{ij}∇_{i}H∇_{j}H

− 2k(k − 2)H(3|A|^{2} − H^{2})|∇H|^{2}

+ 4kλ_{1}λ_{2}H|∇(h_{11} − h_{22})|^{2}

− 8k^{2}λ_{1}λ_{2}(λ_{1} − λ_{2})⟨∇H, ∇(h_{11} − h_{22})⟩

= ( − 4kH|A|^{2}(β^{2} + 3α^{2}) + 4k(k − 1)|A|^{2}(λ_{1}(α + β)^{2})

− 2k(k − 2)H(3|A|^{2} − H^{2})(α + β)^{2} + 4kλ_{1}λ_{2}H(α − β)^{2}

+ 8k^{2}λ_{1}λ_{2}(λ_{1} − λ_{2})(α^{2} − β^{2}))\frac{(∇_{1}h_{11})^{2}}{β^{2}}

+ ( ⋯)\frac{(∇_{2}h_{22})^{2}}{α^{2}}

= − 4λ_{2}^{2}H((3k − 1)λ_{1}^{6} + 4k(1 − k)λ_{1}^{5}λ_{2}

+ (8k^{2} − 9k + 7)λ_{1}^{4}λ_{2}^{2} + 4(1 − k^{2})λ_{1}^{3}λ_{2}^{3}

+ (5k + 1)λ_{1}^{2}λ_{2}^{4} + 4(1 − k)λ_{1}λ_{2}^{5}

+ (k + 1)λ_{2}^{6})\frac{(∇_{1}h_{11})^{2}}{β^{2}} + ( ⋯)\frac{(∇_{2}h_{22})^{2}}{α^{2}}

where the term in brackets in front of (∇_{2}h_{22})^{2}/α^{2} is similar to the one before, just with α and β as well as λ_{1} and λ_{2} interchanged. To complete the proof of the lemma it remains to check that the polynomial in front of (∇_{1}h_{11})^{2}/β^{2} is nonpositive. Since it is homogeneous in λ_{1}, λ_{2} we can choose λ_{2} = 1 and it remains to show that

\begin{aligned}(3k − 1)λ_{1}^{6} & + 4k(1 − k)λ_{1}^{5} + (8k^{2} − 9k + 7)λ_{1}^{4} + 4(1 − k^{2})λ_{1}^{3} \\ & + (5k + 1)λ_{1}^{2} + 4(1 − k)λ_{1} + (k + 1) ⩾ 0 \end{aligned} ; (10)

for λ_{1} ⩾ 0 and 1 ⩽ k ⩽ 5. The inequality for k = 1 is trivial, in the case k = 5 we use a computer algebra system and Sturm’s theorem. One can check that this even extends to k = 5.17... . To see that the inequality holds also for 1 < k < 5 we reorder the polynomial in powers of k:

−(4λ_{1}^{3}(λ_{1}−1)^{2})k^{2}+(3λ_{1}^{6}+4λ_{1}^{5}−9λ_{1}^{4}+5λ_{1}^{2}−4λ_{1}+1)k−λ_{1}^{6}+7λ_{1}^{4}+4λ_{1}^{3}+λ_{1}^{2}+4λ_{1}+1 .

Thus for fixed λ_{1} ⩾ 0 this is a concave function in k, which proves (10). In the case that β = 0 it follows that α ≠ 0 since the evolving surfaces are strictly convex. Equation (9) then yields ∇_{1}h_{11} = ∇_{2}h_{11} = 0. In a similar manner as before we obtain now polynomials in λ_{1}, λ_{2} multiplying (∇_{1}h_{22})^{2} and (∇_{2}h_{22})^{2}. The term in front of (∇_{2}h_{22})^{2} is the same as above and the one in front of (∇_{1}h_{22})^{2} is given by

−12kH|A|^{2}+4k(k−1)λ_{1}|A|^{2}−2k(k−2)H(3|A|^{2}−H^{2})+4kλ_{1}λ_{2}H+8k^{2}λ_{1}λ_{2}(λ_{1}−λ_{2}) .

Choosing λ_{2} = 1 it remains again to check that

−8kλ_{1}^{3} + 8k(k − 1)λ_{1}^{2} − 4k(k + 3)λ_{1} − 4k(k + 1) ⩽ 0

for all λ_{1} ⩾ 0 and 1 ⩽ k ⩽ 5. This can be done as before. The case α = 0 is a mirror image of the case β = 0. □

Proof of Theorem A.1. By Theorem 1.1 the surfaces M_{t} become immediately strictly convex for t > 0. Now choose a sufficiently small 0 < ε < T such that the flow is smooth and strictly convex on the interval (ε, T), where T is the maximal time of existence. Thus the quantity w, as defined above, is well defined on this interval, and bounded from above by Lemma A.2. Note that this implies

1 ⩽ \frac{λ_{max}(p, t)}{λ_{min}(p, t)} ⩽ 1 + \frac{C}{H^{k−1}}

for all (p, t) ∈ M^{2} × (ε, T). This replaces the estimate (3) for k > 1 and the proof follows analogously to the proof of Theorem 1.2. An alternative way of proving Theorem A.1 would be to follow the proofs in [1] or [8]. □

References

[1] Ben Andrews, Contraction of convex hypersurfaces in Euclidian space, Calc. Var. 2 (1994), 151–171.

[2] Ben Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161.

[3] Ben Andrews, Moving surfaces by non-concave curvature functions, 2004,

   arXiv:math.DG/0402273.

[4] Bennett Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Diff. Geom. 22 (1985), 117–138.

[5] Bennett Chow, Deforming hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63–82.

[6] Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22.

[7] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom. 20 (1984), 237–266.

[8] Oliver C. Schnürer, Surfaces contracting with speed |A|^{2}, J. Differential Geom. 71 (2005), no. 3, 347–363.

[9] Felix Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Mathematische Zeitschrift 251 (2005), no. 4, 721–733.

Felix Schulze, FU Berlin, Arnimallee 2-6, 14195 Berlin, Germany

E-mail address: Felix.Schulze@math.fu-berlin.de

Oliver C. Schnürer, FU Berlin, Arnimallee 2-6, 14195 Berlin, Germany

E-mail address: Oliver.Schnuerer@math.fu-berlin.de