New York Journal of Mathematics

New York J. Math. 3 (1997) 103–118.

Harnack Inequalities For Curvature Flows Depending On Mean Curvature

Knut Smoczyk

Abstract. We prove Harnack inequalities for parabolic flows of compact orientable hypersurfaces in ℝ^{n+1}, where the normal velocity is given by a smooth function f depending only on the mean curvature. We use these estimates to prove longtime existence of solutions in some highly nonlinear cases. In addition we prove that compact selfsimilar solutions with constant mean curvature must be spheres and that compact selfsimilar solutions with nonconstant mean curvature can only occur in the case, where f = Aαx^{α} with two constants A and α.

Contents

1. Introduction  103
2. Some relations and evolution equations  106
3. Homothetic solutions  107
4. The Harnack inequality  107
5. Longtime existence for some highly nonlinear flows  114
References  118

1. Introduction

Assume that M^{n} is a compact orientable surface, smoothly immersed into ℝ^{n+1} by a smooth family of diffeomorphisms F_{t} : M^{n} → ℝ^{n+1} that satisfy the PDE

\frac{∂}{∂t}F = −fν ,

where ν denotes the outward pointing unit normal and f is a smooth function depending only on the mean curvature H of the immersed surface, e.g. for f = H we get the well-known mean curvature flow (MCF) and for f = −\frac{1}{H} we obtain the inverse mean curvature flow. Hamilton [4] proved a beautiful Harnack inequality for the MCF. In [1] Harnack inequalities were derived for convex hypersurfaces in cases where f may depend on the full second fundamental form. The case f = −\frac{1}{H}

\begin{matrix}c \\ ◯ \end{matrix}1997 State University of New York ISSN 1076-9803/97

Received September 22, 1997.

Mathematics Subject Classification. 53C21 (35K15).

Key words and phrases. Harnack, mean, curvature, flow, selfsimilar.

has not been studied yet nor are there results for arbitrary functions f depending only on H. The aim of this paper is to address the following questions:

I. What conditions on f guarantee that the flow becomes parabolic so that

we have shorttime existence of a solution? II. For which f do we get nice Harnack inequalities? III. Since selfsimilar solutions play an important role in the Harnack inequality

for the MCF, we ask: For which f do selfsimilar solutions exist and can we say something about the nature of these solutions?

An interesting case is the inverse mean curvature flow f = −\frac{1}{H} since it is important in General Relativity [6]. There is some hope that one can generalize our results to other target manifolds N^{n+1}.

Throughout this paper we will use the standard terminology, i.e., ⟨·, ·⟩ denotes the euclidean inner product, g_{ij}dx^{i}⊗dx^{j} is the induced metric on M^{n}, x^{i} coordinates for M^{n} and h_{ij}dx^{i} ⊗dx^{j} = ⟨∇_{i}F, ∇_{j}ν⟩dx^{i} ⊗dx^{j} denotes the second fundamental form, ∇ the covariant derivative with respect to g_{ij}. Double latin indices are summed from 1 to n and we set

|A|^{2} := h_{ij}h^{ij} , C := h_{ik}h_{l}^{k}h^{il} .

In this paper we will always assume that f : Ω → ℝ is a smooth function defined on an open subset in ℝ and we define

Definition. Let F : M^{n} → ℝ^{n+1} be an immersion. F is called admissible, if H(x) ∈ Ω, ∀x ∈ M^{n}.

The answer to question I is then given by

Proposition I. Let F_{0} : M^{n} → ℝ^{n+1} be an admissible smooth immersion of a compact orientable surface M^{n} and assume that f′ : Ω → ℝ is strictly positive. Then the PDE

(⋆)  \frac{∂}{∂t}F = −fν
F (x, 0) = F_{0}(x), ∀x ∈ M^{n}

has a smooth admissible solution on a maximal time interval [0, T), T > 0.

Proof. This follows from the fact that the linearization of (⋆) differs from the linearization for the mean curvature flow only by a factor f′ which by assumption is strictly positive. Therefore (⋆) is a (nonlinear) parabolic equation and the compactness of M^{n} and the theory for parabolic equations imply shorttime existence. □

In view of Proposition I we will always assume that f′ > 0. Then the main theorem can be stated as follows

Theorem 1. Assume that F_{0} : M^{n} → ℝ^{n+1} is an admissible smooth and convex immersion of an orientable compact M^{n} and that f : Ω → ℝ is a smooth function such that for all x ∈ Ω we have

f′ > 0, \frac{f′′}{f′} x^{2} ≥ ax, (\frac{f′′}{f′} x)′ ≤ 0, ff′′x + ff′ − (f′)^{2}x ≥ 0

where a ∈ ℝ is a constant. Then we can find a small positive constant d such that

\frac{∂}{∂t}f + 2⟨∇f, V ⟩ + h_{ij}V^{i}V^{j} + cf′H ≥ 0

holds for all tangent vectors V as long as t < T, d + (a + 2)t > 0 and M_{t} stays convex, where we have set c(t) := \frac{1}{d+(a+2)t}

Remark. We do have to make these assumptions on f to avoid negative terms in the evolution equation for the basic Harnack expression Z (see below). f = αx^{α} satisfies the assumptions in Theorem 1 on Ω = (0, ∞) with a = α−1. The following Propositions show that almost all functions satisfying the assumptions in Theorem 1 are of this form.

Proposition IIa. Assume that f : (0, a) → ℝ , 0 < a ≤ ∞ is a smooth function that smoothly extends to x = 0 and satisfies all assumptions in Theorem 1. Let us set i_{f} := min{l ≥ 0|f^{(l)}(0) ≠ 0} ≤ ∞. If 0 < i_{f} < ∞ then f = Ai_{f}x^{i_{f}} with a positive constant A.

Proof. Since f(0) = 0 and f′(x) > 0 , ∀x ∈ Ω = (0, a) we observe that f(x) > 0 , ∀x ∈ Ω. By de l’Hospital’s rule we obtain

lim_{x→0} \frac{f′}{f} x = i_{f}

and

lim_{x→0} \frac{f′′}{f′} x = i_{f} − 1

and then (\frac{f′′}{f′} x)′ ≤ 0 and f′ ≥ 0 imply

(1.1)  f′′x ≤ (i_{f} − 1)f′ , ∀x

Since f_{|Ω} > 0 we have by assumption

ff′′x + ff′ − (f′)^{2}x = f^{2}(\frac{f′}{f} x)′ ≥ 0

and then also

(1.2)  f′x ≥ i_{f}f , ∀x

But (1.1) and (1.2) imply

0 ≤ ff′′x + ff′ − (f′)^{2}x ≤ (i_{f} − 1)ff′ + ff′ − i_{f}ff′ = 0

and therefore

(\frac{f′}{f} x)′ = 0

which implies that

\frac{f′}{f} x = i_{f}

and after integration

f = Ai_{f}x^{i_{f}}

with a positive constant A (since f′ > 0). □

Proposition IIb. Assume that f : (a, ∞) → ℝ , 0 < a < ∞ is a smooth function that satisfies all assumptions in Theorem 1 and that g(x) := −f(\frac{1}{x}) smoothly extends to x = 0. If 0 < i_{g} < ∞ then f = −Ai_{g}x^{−i_{g}} with a positive constant A.

Proof. If we set y = \frac{1}{x} and g(y) = −f(x) then the assumptions for f and x translate into the same assumptions for g and y and as above we can derive the desired result. □

Remark. f(x) = \sqrt{x} does not satisfy the assumptions in Proposition IIa since it does not extend smoothly to x = 0. f(x) = x + 1 satisfies all assumptions in Theorem 1. f(x) = ln x satisfies almost all assumptions in Theorem 1. The only condition which is violated is that ff′′x + ff′ − (f′)^{2}x = −\frac{1}{x} < 0 on Ω = (0, ∞).

2. Some relations and evolution equations

By assumption we have

(2.1)  \frac{∂}{∂t}F = −fν

In [7] we formally derived the evolution equations for various geometric objects on M^{n}, these are:

\frac{∂}{∂t}g_{ij} = −2fh_{ij} (2.2)

\frac{∂}{∂t}dμ = −Hfdμ (2.3)

(dμ denotes the volume form)

\frac{∂}{∂t}ν = ∇f (2.4)

\frac{∂}{∂t}h_{ij} = ∇_{i}∇_{j}f − fh_{i}^{l}h_{lj} (2.5)

\frac{∂}{∂t}H = Δf + f|A|^{2} (2.6)

\frac{∂}{∂t}|A|^{2} = 2⟨h_{ij}, ∇_{i}∇_{j}f⟩ + 2fC (2.7)

In addition we have the well-known Gauß-Weingarten-Codazzi-Mainardi equations

∇_{i}∇_{j}F = −h_{ij}ν (2.8)

ΔF = g^{ij}∇_{i}∇_{j}F = −Hν (2.9) ∇_{i}h_{jk} = ∇_{j}h_{ik} (2.10)

∇_{i}ν = h_{i}^{l}∇_{l}F (2.11)

∇_{i}∇_{j}ν = ∇^{l}h_{ij}∇_{l}F − h_{i}^{l}h_{lj}ν (2.12)

Δν = ∇H − |A|^{2}ν (2.13)

Note that in these equations we assume that F, ν are sets of n + 1 functions on M^{n}.

We also have the Simons identity

(2.14)  ∇_{i}∇_{j}H = Δh_{ij} − Hh_{i}^{l}h_{lj} + |A|^{2}h_{ij}

3. Homothetic solutions

A homothetic solution F_{t} is a family of diffeomorphisms such that the surfaces given by the rescaled diffeomorphisms F̃_{t} := ΨF are stationary in ℝ^{n+1}, where Ψ denotes a function depending only on time t. The assumption that F̃_{t} represents a stationary surface means that the normal velocity must be zero. So we have

0 = ⟨\frac{∂}{∂t}F̃, ν̃⟩ = \frac{∂}{∂t}Ψ⟨F, ν⟩ − fΨ

Let us define c := −\frac{∂}{∂t} ln Ψ. Then we have shown that for a homothetic solution of (⋆) we have

(H.1)  f = −c⟨F, ν⟩

Taking covariant derivatives of f and using (2.11), (2.10), (2.8) we obtain with Ṽ_{i} := c⟨F, ∇_{i}F ⟩

∇_{i}f = −h_{i}^{l}Ṽ_{l} (H.2)

∇_{i}Ṽ_{j} = cg_{ij} + fh_{ij} (H.3)

∇_{i}∇_{j}f = −∇^{l}h_{ij}Ṽ_{l} − ch_{ij} − fh_{i}^{l}h_{lj} (H.4)

Δf + f|A|^{2} = −⟨∇H, Ṽ⟩ − cH (H.5)

ΔṼ_{j} = f∇_{j}H + h_{ji}∇^{i}f (H.6)

4. The Harnack inequality

In the sequel we have to calculate many evolution equations. To avoid too complicated formulas it is most convenient to work with coordinates associated to a moving frame. We use similar moving frame coordinates as in [4]. Here the moving frame {E_{a}}_{a=1,...,n} evolves according to

(4.1)  \frac{∂}{∂t}E_{a}^{i} = fh_{j}^{i}E_{a}^{j}

and we denote the coordinates of a vector V with respect to the moving frame by y_{a}^{i}. The following calculations closely follow the procedure in [4]. As in this paper we have

∇_{a}^{b} := y_{a}^{i} \frac{∂}{∂y_{i}^{b}} (4.2)

D_{a} := y_{a}^{i}(\frac{∂}{∂x^{i}} − Γ_{ij}^{k}y_{b}^{j} \frac{∂}{∂y_{b}^{k}}) (4.3)

δ_{ab} := g_{ac}∇_{b}^{c} − g_{bc}∇_{a}^{c} (4.4)

In addition we define

D_{t} := \frac{∂}{∂t} + fh_{a}^{b}∇_{b}^{a} (4.5)

Then straightforward computations give the commutator relations

[D_{a}, D_{b}] = R_{eba}^{d}∇_{d}^{e} (4.6) [∇_{b}^{a}, D^{c}] = −I_{b}^{c}D^{a} ; [∇_{b}^{a}, D_{c}] = I_{c}^{a}D_{b} (4.7) [δ_{bc}, D^{a}] = I_{b}^{a}D_{c} − I_{c}^{a}D_{b} (4.8)

[\frac{∂}{∂t} , D_{a}] = (D_{a}(fh_{b}^{d}) + D_{b}(fh_{a}^{d}) − D^{d}(fh_{ab}))∇_{d}^{b} (4.9)

[D_{t}, D_{a}] = D^{b}(fh_{a}^{c})δ_{bc} + fh_{a}^{b}D_{b} (4.10) [Δ, D_{a}] = D^{b}(R_{lab}^{d}∇_{d}^{l}) + R_{lab}^{d}∇_{d}^{l}D^{b} (4.11)

[D_{t} − f′Δ, D_{a}] = f′R_{a}^{bcd}D_{b}δ_{cd} + f′D_{a}h^{bc}h_{b}^{d}δ_{cd} + \frac{f′′}{f′} D_{a}fΔ (4.12)

+(f − f′H)h_{a}^{l}D_{l} + f′h_{a}^{n}h_{n}^{l}D_{l}
[D_{t}, Δ] = (\frac{f}{f′} − H)D_{n}fD^{n} + 2fh_{a}^{b}D_{b}D^{a} + 2h_{an}D^{a}fD^{n} + D^{a}(D^{b}(fh_{a}^{c})δ_{bc}) (4.13)

In the moving frame the evolution equations reduce to

D_{t}g_{ab} = 0 ; g_{ab} = I_{ab} (4.14) D_{t}h_{ab} = D_{a}D_{b}f + fh_{a}^{n}h_{nb} (4.15) D_{t}f = f′(Δf + f|A|^{2}) (4.16)

Let us now define the following tensors, where we assume that V_{a} is an arbitrary tangent vector on M^{n} and c a smooth function to be determined later and only depending on time t.

X_{a} := D_{a}f + h_{a}^{l}V_{l}
Y_{ab} := D_{a}V_{b} − fh_{ab} − cg_{ab}
Z := D_{t}f + 2⟨Df, V ⟩ + h_{ab}V^{a}V^{b} + cf′H
W_{ab} := D_{t}h_{ab} + D^{l}h_{ab}V_{l} + ch_{ab}
W := D_{t}f + ⟨Df, V ⟩ + cf′H

By (H.1)–(H.6) these tensors vanish on homothetic solutions if we take V_{a} = Ṽ_{a} and the induced c. On the other hand we have

D_{t}Ṽ_{a} = \frac{∂}{∂t}(c⟨F, D_{a}F ⟩) + fh_{c}^{d}∇_{d}^{c}Ṽ_{a}

= \frac{∂}{∂t}(lnc)Ṽ_{a} − c⟨F, D_{a}(fν)⟩ + fh_{a}^{d}Ṽ_{d}

= \frac{∂}{∂t}(lnc)Ṽ_{a} − c⟨F, ν⟩D_{a}f

Harnack Inequalities 109

and therefore

(D_{t} − f′Δ)Ṽ_{a} = \frac{∂}{∂t}(ln c)Ṽ_{a} + f′h_{a}^{b}h_{b}^{c}Ṽ_{c} (4.17)

and on a homothetic solution

(D_{t} − f′Δ)Ṽ_{a} = \frac{∂}{∂t}(ln c)Ṽ_{a} − f′h_{ab}D^{b}f (H.7)

In view of (H.7) we define

U_{a} := (D_{t} − f′Δ)V_{a} − \frac{∂}{∂t}(ln c)V_{a} + f′h_{ab}D^{b}f

which also vanishes on a homothetic solution.

We want to calculate (D_{t} − f′Δ)Z. We do this in several steps. Using (4.12) and (4.16) we obtain

(D_{t} − f′Δ)D_{a}f = \frac{f′′}{f′} D_{a}fΔf + (f − f′H)h_{a}^{b}D_{b}f + f′h_{a}^{n}h_{n}^{l}D_{l}f + D_{a}(ff′|A|^{2})

and therefore

(4.18)  (D_{t} − f′Δ)D_{a}f =f′|A|^{2}D_{a}f + 2ff′h^{bc}D_{a}h_{bc} + f′h_{a}^{n}h_{n}^{l}D_{l}f

+ (f − f′H)h_{a}^{b}D_{b}f + \frac{f′′}{(f′)^{2}} D_{a}fD_{t}f

Further, we use Simons identity to rewrite (4.15)

D_{t}h_{ab} = D_{a}D_{b}f + fh_{a}^{n}h_{nb}

= f′(Δh_{ab} − Hh_{a}^{n}h_{nb} + |A|^{2}h_{ab}) + \frac{f′′}{(f′)^{2}} D_{a}fD_{b}f + fh_{a}^{n}h_{nb}

This gives

(4.19)  (D_{t} − f′Δ)h_{ab} = f′|A|^{2}h_{ab} + (f − f′H)h_{a}^{n}h_{nb} + \frac{f′′}{(f′)^{2}} D_{a}fD_{b}f

(4.18), (4.19) and the definition of U give

(D_{t} − f′Δ)X_{a} = f′|A|^{2}D_{a}f + 2ff′h^{bc}D_{a}h_{bc} + f′h_{a}^{n}h_{n}^{l}D_{l}f + (f − f′H)h_{a}^{b}D_{b}f

+ \frac{f′′}{(f′)^{2}} D_{a}fD_{t}f + V^{b}(f′|A|^{2}h_{ab} + (f − f′H)h_{a}^{n}h_{nb} + \frac{f′′}{(f′)^{2}} D_{a}fD_{b}f)

+ h_{a}^{b}(U_{b} + \frac{∂}{∂t}(ln c)V_{b} − f′h_{bc}D^{c}f) − 2f′D^{c}h_{a}^{b}D_{c}V_{b}

= f′|A|^{2}X_{a} + (f − f′H)h_{a}^{b}X_{b} + h_{a}^{b}U_{b} − 2f′D_{a}h_{bc}(Y^{bc} + cg^{bc})

+ \frac{∂}{∂t}(ln c)(X_{a} − D_{a}f) + \frac{f′′}{(f′)^{2}} D_{a}f(W − cf′H)

and finally

(D_{t} − f′Δ)X_{a} =f′|A|^{2}X_{a} + (f − f′H)h_{a}^{b}X_{b} + h_{a}^{b}U_{b} − 2f′D_{a}h_{bc}Y^{bc} (4.20)

+ \frac{∂}{∂t}(ln c)X_{a} + \frac{f′′}{(f′)^{2}} D_{a}fW − (2c + \frac{∂}{∂t}(ln c) + c\frac{f′′}{f′} H)D_{a}f

Next we compute

(D_{t} − f′Δ)D_{t}f =[D_{t}, f′Δ]f + D_{t}(ff′|A|^{2})
=f′[D_{t}, Δ]f + D_{t}f′Δf + D_{t}f′f|A|^{2} + f′|A|^{2}D_{t}f + ff′D_{t}|A|^{2}
=(f − f′H)|Df|^{2} + 2ff′h_{ab}D^{a}D^{b}f + 2f′h_{ab}D^{a}fD^{b}f

+ \frac{f′′}{(f′)^{2}} (D_{t}f)^{2} + f′|A|^{2}D_{t}f + 2ff′h^{ab}D_{t}h_{ab}

giving

(4.21) (D_{t} − f′Δ)D_{t}f =f′|A|^{2}D_{t}f + 4ff′h^{ab}D_{t}h_{ab} − 2f^{2}f′C + \frac{f′′}{(f′)^{2}} (D_{t}f)^{2}
+ (f − f′H)|Df|^{2} + 2f′h_{ab}D^{a}fD^{b}f

Furthermore

(D_{t} − f′Δ)(cf′H) =\frac{∂}{∂t}(ln c)cf′H + c\frac{f′′}{f′} HD_{t}f + cD_{t}f
− cf′(H(f′′ΔH + f′′′|DH|^{2}) + f′ΔH + 2f′′|DH|^{2})
=\frac{∂}{∂t}(ln c)cf′H + c(f′′H + f′)(Δf + f|A|^{2})
− c(f′′H + f′)(f′ΔH) − cf′(Hf′′′ + 2f′′)|DH|^{2}
=\frac{∂}{∂t}(ln c)cf′H + cf(f′ + f′′H)|A|^{2} + c((f′′)^{2}H − f′f′′ − f′f′′′H)|DH|^{2}

which gives

(4.22) (D_{t} − f′Δ)(cf′H) = \frac{∂}{∂t}(ln c)cf′H + cf(f′ + f′′H)|A|^{2} − c(\frac{f′′}{f′} H)′|Df|^{2}

(4.21), (4.22) and (4.18) give

(D_{t} − f′Δ)W =f′|A|^{2}D_{t}f + 4ff′h^{ab}D_{t}h_{ab} − 2f^{2}f′C + \frac{f′′}{(f′)^{2}} (D_{t}f)^{2} + (f − f′H)|Df|^{2}

+ 2f′h_{ab}D^{a}fD^{b}f + \frac{∂}{∂t}(ln c)cf′H + cf(f′ + f′′H)|A|^{2} − c(\frac{f′′}{f′} H)′|Df|^{2}

+ V^{a}(f′|A|^{2}D_{a}f + 2ff′h^{bc}D_{a}h_{bc} + f′h_{a}^{n}h_{n}^{l}D_{l}f

+ (f − f′H)h_{a}^{b}D_{b}f + \frac{f′′}{(f′)^{2}} D_{a}fD_{t}f)

+ D^{a}f(U_{a} + \frac{∂}{∂t}(ln c)V_{a} − f′h_{ab}D^{b}f) − 2f′D^{a}V^{b}D_{a}D_{b}f

=f′|A|^{2}W − c((f′)^{2}H − ff′ − ff′′H)|A|^{2} + \frac{∂}{∂t}(ln c)W − \frac{∂}{∂t}(ln c)D_{t}f

+ \frac{f′′}{(f′)^{2}} D_{t}fW − c\frac{f′′}{f′} HD_{t}f + (f − f′H)⟨X, Df⟩

+ f′(4fh^{ab} − 2D^{a}V^{b})D_{t}h_{ab} + 2ff′D^{a}V^{b}h_{a}^{n}h_{nb} − 2f^{2}f′C

− c(\frac{f′′}{f′} H)′|Df|^{2} + 2ff′V^{a}D_{a}h_{bc}h^{bc} + ⟨Df, U⟩ + f′h_{ab}X^{a}D^{b}f

=f′|A|^{2}W − c((f′)^{2}H − ff′ − ff′′H)|A|^{2} + \frac{∂}{∂t}(ln c)W

− (c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c))D_{t}f + \frac{f′′}{(f′)^{2}} D_{t}fW + (f − f′H)⟨X, Df⟩

+ f′(4fh^{ab} − 2D^{a}V^{b})W_{ab} − f′(4fh^{ab} − 2D^{a}V^{b})(D^{l}h_{ab}V_{l} + ch_{ab})

+ 2ff′D^{a}V^{b}h_{a}^{n}h_{nb} − 2f^{2}f′C − c(\frac{f′′}{f′} H)′|Df|^{2}

+ 2ff′V^{a}D_{a}h_{bc}h^{bc} + f′⟨h_{ab}, D_{a}fX_{b}⟩ + ⟨Df, U⟩

and after rearranging terms we conclude

(D_{t} − f′Δ)W =f′|A|^{2}W + f′(4fh^{ab} − 2D^{a}V^{b})W_{ab} + f′h^{ab}X_{a}D_{b}f + ⟨Df, U⟩ (4.23)

+ 2f′(fh_{a}^{n}h_{nb} + D^{l}h_{ab}V_{l})Y^{ab} + (2c + \frac{∂}{∂t}(ln c))W

− c((f′)^{2}H − ff′ − ff′′H)|A|^{2} − (c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)D_{t}f

+ \frac{f′′}{(f′)^{2}} D_{t}fW + (f − f′H)⟨X, Df⟩ − c(\frac{f′′}{f′} H)′|Df|^{2} + 2cf′h_{ab}Y^{ab}

Eventually the evolution equation for Z is given by

112 Knut Smoczyk

(D_{t} − f′Δ)Z =f′|A|^{2}W + f′(4fh^{ab} − 2D^{a}V^{b})W_{ab} + f′h^{ab}X_{a}D_{b}f + ⟨Df, U⟩

+ 2f′(fh_{a}^{n}h_{nb} + D^{l}h_{ab}V_{l})Y^{ab} + (2c + \frac{∂}{∂t}(ln c))W

− c((f′)^{2}H − ff′ − ff′′H)|A|^{2} − (c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)D_{t}f

+ \frac{f′′}{(f′)^{2}} D_{t}fW + (f − f′H)⟨X, Df⟩ − c(\frac{f′′}{f′} H)′|Df|^{2} + 2cf′h_{ab}Y^{ab}

+ V^{a}(f′|A|^{2}X_{a} + (f − f′H)h_{a}^{b}X_{b} + h_{a}^{b}U_{b} − 2f′D_{a}h_{bc}Y^{bc}

+ \frac{∂}{∂t}(ln c)X_{a} + \frac{f′′}{(f′)^{2}} D_{a}fW − (2c + \frac{∂}{∂t}(ln c) + c\frac{f′′}{f′} H)D_{a}f)

+ X^{a}(U_{a} + \frac{∂}{∂t}(ln c)V_{a} − f′h_{ab}D^{b}f) − 2f′D_{a}V_{b}(W^{ab} + h^{an}Y_{n}^{b})

=f′|A|^{2}Z + 2⟨X, U⟩ − 4f′(Y^{ab} + cg^{ab})W_{ab} − 2f′h^{an}Y_{n}^{b}(Y_{ab} + cg_{ab})

+ (f − f′H)|X|^{2} + \frac{f′′}{(f′)^{2}} W (D_{t}f + ⟨V, Df⟩) − (c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)(D_{t}f + ⟨Df, V ⟩)

+ 2\frac{∂}{∂t}(ln c)⟨X, V ⟩ + (2c + \frac{∂}{∂t}(ln c))W − c((f′)^{2}H − ff′ − ff′′H)|A|^{2}

− c(\frac{f′′}{f′} H)′|Df|^{2} + 2cf′h_{ab}Y^{ab}

=f′|A|^{2}Z + 2⟨X, U⟩ − 2f′h_{a}^{n}Y_{nb}Y^{ab} − 4f′W_{ab}Y^{ab} − 4cW

+ (f − f′H)|X|^{2} + \frac{f′′}{(f′)^{2}} W (W − cf′H) − (c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)(W − cf′H)

+ 2\frac{∂}{∂t}(ln c)⟨X, V ⟩ + (2c + \frac{∂}{∂t}(ln c))W − c((f′)^{2}H − ff′ − ff′′H)|A|^{2} − c(\frac{f′′}{f′} H)′|Df|^{2}

So we finally arrive at

(D_{t}−f′Δ)Z = (f′|A|^{2} + \frac{f′′}{(f′)^{2}} (Z − 2⟨X, V ⟩) − 2(c\frac{f′′}{f′} H + 2c))Z (4.24)

+ 2⟨X, U⟩ + (f − f′H)|X|^{2} + \frac{f′′}{(f′)^{2}} ⟨X, V ⟩^{2} + 2(c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)⟨X, V ⟩

− 2f′h_{a}^{n}Y_{nb}Y^{ab} − 4f′W_{ab}Y^{ab} + (c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)cf′H

+ c(ff′′H + ff′ − (f′)^{2}H)|A|^{2} − c(\frac{f′′}{f′} H)′|Df|^{2}

Now we are able to prove Theorem 1. Proof of Theorem 1. From (4.24) and the assumptions in Theorem 1 we conclude, with c := \frac{1}{d+(a+2)t}

(D_{t} − f′Δ)Z ≥(f′|A|^{2} + \frac{f′′}{(f′)^{2}} (Z − 2⟨X, V ⟩) − 2(c\frac{f′′}{f′} H + 2c))Z − 2f′h_{a}^{n}Y_{nb}Y^{ab} − 4f′W_{ab}Y^{ab}

+ 2⟨X, U⟩ + (f − f′H)|X|^{2} + \frac{f′′}{(f′)^{2}} ⟨X, V ⟩^{2} + 2(c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)⟨X, V ⟩

Now choose d so small such that Z > ϵ > 0 for t = 0 and for all tangent vectors V. This is possible since the initial surface is convex. On any compact time interval [0, t_{0}] with t_{0} < T we can therefore estimate

(D_{t}−f′Δ)Z ≥ −bZ − 2f′h_{a}^{n}Y_{nb}Y^{ab} − 4f′W_{ab}Y^{ab}

+ 2⟨X, U⟩ + (f − f′H)|X|^{2} + \frac{f′′}{(f′)^{2}} ⟨X, V ⟩^{2} + 2(c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)⟨X, V ⟩

with a large constant b depending on t_{0}. Additionally for t > 0 and an arbitrary positive constant δ

(D_{t} − f′Δ)(e^{bt}Z + δt) >e^{bt}(−2f′h_{a}^{n}Y_{nb}Y^{ab} − 4f′W_{ab}Y^{ab} + 2⟨X, U⟩ + (f − f′H)|X|^{2}

+ \frac{f′′}{(f′)^{2}} ⟨X, V ⟩^{2} + 2(c\frac{f′′}{f′} H + \frac{∂}{∂t}(ln c) + 2c)⟨X, V ⟩)

If t_{1} ≤ t_{0} would be the first time where e^{bt}Z + δt would become zero at some point x ∈ M^{n} and for some tangent vector V then we must have X_{a} = 0 since it is the first variation with respect to V and we can also extend V in spacetime such that Y_{ab} = 0. This implies a contradiction and therefore e^{bt}Z + δt > 0, ∀t ≤ t_{0}. Since δ and t_{0} are arbitrary we conclude Z ≥ 0 whenever t < T and d + (a + 2)t > 0. □

Now we want to answer the question for which f one can expect selfsimilar solutions of (⋆). First we remark that if x ∈ Ω and x > 0, then the sphere of constant radius \frac{n}{x} and with constant mean curvature H = x gives always rise to a selfsimilar solution for a short time.

For any subset A ⊂ M^{n} let us define

H(A) := {x ∈ Ω|∃p ∈ A : H(p) = x}

and

P_{t} := {p ∈ M_{t}^{n}|∇H ≠ 0}

The answer to question III is then given by

Proposition IIIa. If P_{0} ≠ ∅ and F_{t} : M^{n} → ℝ^{n+1} is a selfsimilar solution of (⋆) for a compact connected M^{n}, then we have f = Aαx^{α} , ∀x ∈ H(M_{t}) with nonvanishing constants A and α.

Proof. Since F_{t} is selfsimilar we have P_{t} = P_{0} , ∀t ∈ [0, T). Since X_{a}, W vanish on selfsimilar solutions so must their time derivatives. From (4.20) we conclude that on H(P_{t}) and by continuity also on \overline{H(P_{t})} we must have c\frac{f′′}{f′} x + 2c + \frac{∂}{∂t}(ln c) = 0. Since P_{0} ≠ ∅ and M^{n} is connected we have H(M^{n}) ⊂ \overline{H(P_{t})} and we derive

c\frac{f′′}{f′} x + 2c + \frac{∂}{∂t}(ln c) = 0

for all x ∈ H(M_{t}^{n}). Since f′ > 0 there can be at most one point x ∈ Ω such that f(x) = 0. At all other points we have

ff′′x + ff′ − (f′)^{2}x = f^{2}(\frac{f′}{f} x)′

Then (4.23) implies that also

(\frac{f′}{f} x)′ = 0

in all points x ∈ H(M_{t}) where f(x) ≠ 0. But after integration we obtain that at these points

f = Aαx^{α}

with constants A, α (nonvanishing since we must have f′ > 0) and again by continuity this also holds on all of H(M_{t}). □

We observe that by (2.9)

Δ|F |^{2} = 2(n − H⟨F, ν⟩)

and with (H.1) we obtain on a homothetic solution

(H.8)  Δ|F |^{2} = 2(n − \frac{fH}{c})

This implies

Proposition IIIb. If P_{0} = ∅ and M^{n} is a compact orientable selfsimilar solution of (⋆), then M_{t} is a sphere of radius \frac{n}{H}. If f = −\frac{1}{H}, then any compact orientable homothetic solution is a sphere of radius \frac{n}{H}.

Proof. P_{0} = ∅ and (H.8) imply Δ|F |^{2} = const. By the assumptions on M^{n} this constant must be zero and then consequently |F |^{2} = const. This implies that M_{t} is a sphere. □

5. Longtime existence for some highly nonlinear flows

In this paragraph we are going to show that one can use the Harnack inequality to prove longtime existence of solutions for some highly nonlinear flows. To be precise

Theorem 2. Assume that f : (0, ∞) → ℝ is a smooth negative function that satisfies the assumptions in Theorem 1 with a > −2 and that lim_{x→0} f(x) = −∞ and F_{0} : M^{2} → ℝ^{3} is a smooth convex immersion of an orientable compact surface M^{2}. Then (⋆) has a smooth immortal solution and we have

lim_{t→∞} H = 0 , lim_{t→∞} |F_{t}|^{2} = ∞

Remark. The functions f = αx^{α} with −1 < α < 0 satisfy the assumptions in Theorem 2 on Ω = (0, ∞). These speed functions are not homogenous of degree one and are not included in the class of functions considered in [2]

To prove Theorem 2 we need some lemmas that are interesting on their own.

Lemma 1. Assume f′ > 0 and that F_{0} : M^{n} → ℝ^{n+1} is a smooth immersion of an orientable compact surface and that on M_{0} = F_{0}(M^{n}) we have f^{2} ≥ min_{M_{0}} f^{2} > 0. Then this is also true on the maximal time interval [0, T) where a smooth solution of (⋆) exists.

Proof. The evolution equation for f^{2} is given by

(D_{t} − f′Δ)f^{2} = f′(−2|∇f|^{2} + 2f^{2}|A|^{2})

and the result follows from the parabolic maximum principle. □

Corollary 1. With the assumptions in Lemma 1 we have that for negative f

H ≤ max_{M_{0}} H

and for positive f

H ≥ min_{M_{0}} H

We can do even better

Lemma 2. Assume Ω = (0, ∞) , f < 0 , f′ > 0 and that F_{0} : M^{n} → ℝ^{n+1} is an admissible immersion of an orientable compact manifold M^{n}. Then we can find a positive ϵ such that on [0, T)

max_{M_{t}} H ≤ \frac{max_{M_{0}} H}{1 + ϵt max_{M_{0}} H}

Proof. Since F_{t} is admissible on [0, T) we must always have H > 0. At a point where max_{M_{t}} H is attained we have ΔH ≤ 0 and ∇H = 0. The evolution equation for H (2.6), Lemma 1 and the fact that |A|^{2} ≥ \frac{H^{2}}{n} give us

\frac{∂}{∂t} max_{M_{t}} H ≤ −ϵ(max_{M_{t}} H)^{2}

with ϵ = \frac{− max_{M_{0}} f}{n} and after integration we obtain the result. □

(2.7) and Simons identity give

(5.1) (D_{t} − f′Δ)|A|^{2} = −2f′|∇A|^{2} + 2f′|A|^{4} + 2(f − f′H)C + 2f′′h_{ij}∇^{i}H∇^{j}H

Let R = H^{2} − |A|^{2} be the Scalar curvature of M_{t}. Now we turn our attention to the case where n = 2. In this case we can decompose C into

C = \frac{H}{2} (3|A|^{2} − H^{2})

Then (5.1) and (2.6) imply that for n = 2

(5.2) (D_{t} − f′Δ)R =2f′(|∇A|^{2} − |∇H|^{2}) + 2f′′(H|∇H|^{2} − h_{ij}∇^{i}H∇^{j}H)
− (2f′|A|^{2} + (f − f′H)H)R

Lemma 3. Under the assumptions in Theorem 2 we have

R > 0, ∀t ∈ [0, T)

Proof. Let t_{0} ≤ T be the maximal time such that R > 0 on all of M_{t} for t ∈ [0, t_{0}), i.e., the maximal time for which M_{t} stays convex. If t_{0} = T we are done. So assume that t_{0} < T. Since M_{t} is admissible for t ∈ [0, t_{0}] we must have H > 0 on [0, t_{0}]. By definition of t_{0} we also have R > 0 on [0, t_{0}) and R ≥ 0 on [0, t_{0}]. Since M and [0, t_{0}] are both compact and R ≥ 0 we can find a constant c such that

(5.3) (D_{t} − f′Δ)R ≥ 2f′(|∇A|^{2} − |∇H|^{2}) + 2f′′(H|∇H|^{2} − h_{ij}∇^{i}H∇^{j}H) − cR

Now assume that x is any point on M_{t} where the minimum of R is attained. At this point we must have f′ΔR ≥ 0 and ∇R = 0. Let α and β denote the two principal curvatures in a neighborhood of x. We have

0 = ∇R = 2α∇β + 2β∇α

Since H > 0 and α, β ≥ 0 we must either have α > 0 or β > 0. Let us assume that α > 0. First we compute that

H|∇H|^{2} − h_{ij}∇^{i}H∇^{j}H = α|∇_{2}H|^{2} + β|∇_{1}H|^{2} ≥ 0,

where we choose normal coordinates such that h_{ij} = diag(α, β). Since f′′x ≥ af′ and H > 0, a > −2 we can estimate

2f′′(H|∇H|^{2} − h_{ij}∇^{i}H∇^{j}H) ≥ −4\frac{f′}{H} (α|∇_{2}H|^{2} + β|∇_{1}H|^{2})

On the other hand Codazzi’s equation gives us that

|∇A|^{2} = |∇_{1}α|^{2} + |∇_{2}β|^{2} + 3|∇_{2}α|^{2} + 3|∇_{1}β|^{2}

Combining the last two statements we see that at a point where ∇R = 0 we must always have

2f′(|∇A|^{2} − |∇H|^{2}) + 2f′′(H|∇H|^{2} − h_{ij}∇^{i}H∇^{j}H) ≥ 0

and consequently at any point where the minimum of R is attained

D_{t}R ≥ −cR

which implies that

min_{M_{t}} R ≥ (min_{M_{0}} R)e^{−ct} > 0, ∀t ∈ [0, t_{0}]

This proves that t_{0} = T. □

We come to the proof of Theorem 2 Proof of Theorem 2. On [0, T) we must have H > 0: Otherwise (⋆) is not welldefined, i.e., the surfaces would not be admissible. If T is finite, then this can only happen for two reasons. Either the solutions converge to a surface that is no longer admissible, i.e., the mean curvature would vanish somewhere, or the surfaces must develop a singularity. Since M_{t} is convex on [0, T) we can apply Theorem 1 with V = 0 to obtain

\frac{∂}{∂t}H ≥ −\frac{1}{d + (a + 2)t}H

and since by assumption a > −2 we can estimate

\frac{∂}{∂t}H ≥ −\frac{1}{d}H

which implies that

H ≥ e^{\frac{−1}{d} t} min_{M_{0}} H

So H cannot become zero in finite time. This estimate and Corollary 1 imply that the surfaces stay admissible in finite time. Therefore we conclude that if T is finite the surfaces must develop a singularity. It is well-known that the second fundamental form |A|^{2} must then blow up for t → T (compare [2] and [5]). By Lemma 3 we conclude that in the case where T < ∞, H must also blow up which, in view of Corollary 1, proves that T = ∞. By Lemma 2 we conclude that lim_{t→∞}(max_{M_{t}} H) = 0. It remains to prove that |F_{t}| → ∞. An easy calculation gives the following evolution equation

(D_{t} − f′Δ)(\frac{⟨F, ν⟩}{f} + (2 + a)t) = 2\frac{f′}{f} ⟨∇(\frac{⟨F, ν⟩}{f} + (2 + a)t), ∇f⟩ + a + 1 − \frac{f′H}{f}

The assumptions on f imply that lim_{x→0} \frac{f′′x}{f′} exists and that lim_{x→0} \frac{f′′x}{f′} ≥ a.

Since lim_{x→0} f = −∞ we can apply de l’Hospital’s rule and get lim_{x→0} \frac{f′x}{f} =

1 + lim_{x→0} \frac{f′′x}{f′} ≥ a + 1. On the other hand the assumption that (\frac{f′x}{f})′ ≥ 0 implies for any x ∈ Ω

a + 1 − \frac{f′x}{f} ≤ 0

This gives

(D_{t} − f′Δ)(\frac{⟨F, ν⟩}{f} + (2 + a)t) ≤ 2\frac{f′}{f} ⟨∇(\frac{⟨F, ν⟩}{f} + (2 + a)t), ∇f⟩

and with the maximum principle we conclude

\frac{⟨F, ν⟩}{f} ≤ c − (2 + a)t

for a constant c. Using Schwarz’ inequality we have proven that

\frac{|F |}{|f|} ≥ (2 + a)t − c

and since lim_{t→∞} |f| → ∞ and 2+a > 0, this can only hold when lim_{t→∞} |F | = ∞, proving the rest of Theorem 2. □

118 Knut Smoczyk

References

[1] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Zeitschrift 217 (1994), 179–197.

[2] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J.D.G 32 (1990), 299–314.

[3] R.S. Hamilton, The Harnack estimate for the Ricci Flow, J.D.G 37 (1993), 225–243.

[4] R.S. Hamilton, Harnack estimate for the mean curvature flow, J.D.G 41 (1995), 215–226.

[5] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J.D.G 20 (1984), 237– 268.

[6] G. Huisken, T. Ilmanen, Proof of the Penrose inequality, to appear.

[7] K. Smoczyk, Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature, Calc. Var. 4 (1996), 155–170.

[8] S.T. Yau, On the Harnack inequalities of partial differential equations, Comm. Anal. Geom. 2 (1994), 431–430.

ETH Zürich, Math. Department, CH-8092 Zürich, Switzerland ksmoczyk@math.ethz.ch

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