Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
Abstract.
We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form , where and is a positive, strictly monotone and homogeneous curvature function. In particular this class includes the mean curvature . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to AndrewsMcCoyZheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.
Key words and phrases:
Inverse curvature flow; Pinching2010 Mathematics Subject Classification:
53C21, 53C441. Introduction
We consider inverse curvature flows in either the Euclidean space or the hyperbolic space with sectional curvature , i.e. a family of embeddings
(1.1) 
where is a closed, connected and orientable manifold of dimension , which solves
(1.2) 
Here is the outward pointing normal to the flow hypersurface and is evaluated at the principal curvatures at the point .
Let us first state our main theorem. Therefore we will use the following assumptions for the curvature function and the initial hypersurface .
1.1 Assumption.
Let and
(1.3) 
Let be a positive, strictly monotone and symmetric curvature function which is homogeneous of degree one and normalized to .
1.2 Assumption.
Let satisfy Assumption 1.1. Let
(1.4) 
be the smooth embedding of a hypersurface with which can be written as a graph over a geodesic sphere ,
(1.5) 
In the following statement of our main theorem, we let
(1.6) 
be the mean curvature, the Weingarten operator, be its norm with respect to the induced metric and the identity. The principal curvatures , , will always be labelled according to
(1.7) 
1.3 Theorem.
Let and let be either the Euclidean space or the hyperbolic space of constant sectional curvature Let and let , satisfy Assumptions 1.2, LABEL: and 1.1. Furthermore, we assume that in case , satisfies the pinching condition
(1.8) 
and in case , that satisfies the pinching condition
(1.9) 
where is sufficiently small. Then:

There exists a unique smooth solution on a maximal time interval
(1.10) of the equation
(1.11) where in case and in case , where is the outward unit normal to at and is evaluated at the principal curvatures of at .

The flow hypersurfaces can be written as graphs of a function over so that
(1.12) and properly rescaled flow hypersurfaces converge for all in to a geodesic sphere.

In case there exists a point and a sphere around with radius such that the spherical solutions with radii of (1.11) with satisfy
(1.13) . Here denotes the Hausdorff distance of compact sets.
1.4 Remark.
(i) The rescalings, mentioned in (ii) of this theorem, are given by
(1.14) 
in case and by
(1.15) 
in case where is the unique radius of a sphere which exists exactly as long as the flow, i.e. for the time
(ii) A result similar to Theorem 1.3 (iii) can not be deduced in the hyperbolic space, cf. the nice counterexample in [22].
(iii) For flows by high powers of curvature, pinching conditions similar to (1.9) and (1.11) have already appeared for contracting flows in [3] and [39]. Indeed we can mimic a counterexample to preserved convexity for contracting flows by AndrewsMcCoyZheng [4, Thm. 3] and show that in general strict convexity (in particular general pinching) will be lost if and .
Theorem 1.3 shows the following: If we remove the assumptions , as well as the concavity of in [17, Theorem 1.2] and [34, Theorem 1.2] and replace them by a certain pinching condition for the initial hypersurface then the resulting theorems are true. This allows us e.g. to consider the interesting case and more generally
(1.16) 
where is the th elementary symmetric polynomial.
Expanding curvature flows of the form (1.2) with have attracted a lot of attention, since they have proven to be useful in the deduction and generalization of several geometric inequalities, like the Riemannian Penrose inequality, [21], some Minkowski type inequalities, [5] and [20], and AlexandrovFenchel type inequalities as in [12]. The case has been treated in broad generality, cf. [13] and [40, 41] for the Euclidean case, [16] for the hyperbolic case and [6, 11, 18, 23, 25, 28, 29, 36, 43] for other ambient spaces. For the case there are fewer results. In the Euclidean case there is [17, 26, 38] and in the hyperbolic case there is [34]. All of these papers share the fact that only curvature functions are considered which vanish on the boundary of , a property which forces preservation of convexity by definition. The goal of the present paper is the generalisation of both of these works. The second author already obtained an improvement of [34] in [33], where he could drop the pinching assumption on the initial hypersurface for some powers of the inverse Gauss curvature flow. But in particular for powers of the inverse mean curvature flow there are no results, up to the authors’ knowledge. In the recent preprint [24] Li, Wang and Wei proved convergence results for the case in and . The novelty in this paper is that they could also treat nonconcave curvature functions, since for parabolic equations in two variables one can replace the KrylovSafonov estimates by a regularity result due to Andrews [1]. Wei provided some new pinching estimates in the cases for a broad class of curvature functions in [42]. Some nonhomogeneous flow speeds are considered in [8] and [9].
The paper is organised as follows. In sections 3 and 2 we collect some notation and evolution equations. In section 4 we give the counterexample. In section 5 we prove the crucial preservation of a specific pinching of the initial hypersurface, whereafter in sections 7 and 6 we give an outline on how to finish the proof of Theorem 1.3. Here one could basically follow the lines of [17] and [34], but due to the pinching estimates several aspects of the proofs in these references simplify so we present these simplified arguments for convenience.
2. Notation and preliminaries
Let and be either the Euclidean space or the hyperbolic space with constant sectional curvature of dimension . Let be a compact, connected, smooth manifold and
(2.1) 
be an embedding with unit outward normal vector field (compare the nice note [32]). Let be the induced metric on , where are the components of with respect to the basis , . In tensor expressions latin indices always range between and and greek indices range from to indicating components of tensors of the ambient space. The coordinate expression of a covariant derivative with respect to the LeviCivita connection of of a tensor field are indicated by a semicolon,
(2.2) 
The second fundamental form is given by the Gaussian formula
(2.3) 
and the Weingarten map is denoted by .
For any the pointed Euclidean as well as the hyperbolic space is diffeomorphic to and is covered by geodesic polar coordinates. The metric is given by
(2.4) 
where is the geodesic distance to , is the round metric on and
(2.5) 
The principal curvatures of the coordinates slices are given by
(2.6) 
Since our hypersurfaces will all be convex, they can be written as graphs in geodesic polar coordinates over
(2.7) 
where is a smooth function on Define
(2.8) 
In terms of a graph, the second fundamental form of can be expressed as
(2.9) 
cf. [15, Rem. 1.5.1].
Let us also make some comments on the speed functions under which the family of embeddings evolve. By Assumption 1.1 these are given by smooth, symmetric functions on an open, symmetric and convex cone . It is well known, that such a function can be written as a smooth function of the elementary symmetric polynomials ,
(2.10) 
compare [19]. The associated functions to , which are defined on endomorphisms of the tangent space, are traditionally denoted by and given by
(2.11) 
for all , cf. [15, equ. (2.1.31)]. Hence also can be viewed as a function defined on the endomorphism bundle , i.e.
(2.12) 
We will, however, mostly use a different description of , namely as being defined on two variables by setting
(2.13) 
where is the inverse of the positive definite tensor and . We denote by and the first and second derivatives of with respect to , i.e.
(2.14) 
Due to the monotonicity assumption on as a function of the principal curvatures, the tensor is positive definite at all pairs , such that has eigenvalues in . Compare [2], [15, Ch. 2] and [37] for more details.
We also note that we will in the sequel use the same symbol for both functions and . This will not cause confusion, since in expressions like it only makes sense to think of .
We always assume that , set for and write
(2.15) 
3. Evolution equations
The proof of the following evolution equations is given in [15, Lemma 2.3.4], [15, Lemma 3.3.2] and [14, Lemma 5.8].
3.1 Lemma.
Under the flow (1.11) the geometric quantities
(3.1) 
and evolve as follows
(3.2) 
(3.3) 
and
(3.4) 
where denotes the mean curvature of the coordinate slice
3.2 Lemma.
Under the flow (1.11) the Weingarten map form evolves by
(3.5)  
For the tensor
(3.6) 
evolves by
(3.7)  
and hence evolves by
(3.8)  
4. A counterexample to preserved convexity
For contracting flows, i.e. flows of the form
(4.1) 
with positive , AndrewsMcCoyZheng [4] gave an example of a (weakly) convex hypersurface in which develops a negative principal curvature instantly for a certain class of speeds . By continuity with respect to initial values this shows that for these one can also find strictly convex initial hypersurfaces which develop negative principal curvatures quickly.
In this section we briefly sketch that we generally face the same phenomenon in our case of expanding flows. Precisely we will see that for powers of the inverse mean curvature flow convexity might be lost, indicating that a stronger pinching condition as for example in Theorem 1.3 is needed.
We will show that the loss of convexity can occur along the flow
(4.2) 
for simplicity in . Contrary to the contracting case, where a more sophisticated speed is needed, here we can make use of the strong concavity of the function , which gives an additional negative term in
(4.3) 
Let us briefly recall the method in [4, Thm. 3] how to construct such a convex initial hypersurface. First they construct a local graph using the function
(4.4) 
where are arbitrary positive numbers and
(4.5) 
There holds and hence at , compare [4, equ. (16),(17),(18)],
(4.6) 
(4.7) 
and
(4.8) 
where indices appearing after a comma denote usual partial derivatives. In the proof of [4, Thm. 2] it is shown that the graph of over a small ball is a convex hypersurface, which is strictly convex in , and that this graph can be closed up to a convex hypersurface, respecting the strict convexity in . Due to , the hypersurface is strictly mean convex and (4.2) is defined for short time.
It remains to show that under the flow (4.2), the entry of the second fundamental form, which equals zero at , drops below zero instantly. According to (3.5) we have at :
(4.9)  
for a suitable arrangement of and , due to . Hence drops below zero instantly and the example is complete.
5. The pinching estimates
We define where or . We set and in both cases.
5.1 Lemma.
Proof.
While in the Euclidean case the proof is similar to the proof of [35, Prop. 3.4] we need further arguments in the hyperbolic case. We begin with some facts which hold in both cases. Let and define the quantity
(5.2) 
Then, due to Lemma 3.2, satisfies the evolution equation
(5.3)  
and due to [3, Lemma 2.1] we have
(5.4)  
(i) Let us now assume that . In view of (5.1) we have . We want to show that for all . For this we assume that there is a minimal and so that . From [3, Lemma 2.3] we deduce that
(5.5)  
in .
Using (5.3) we conclude that
(5.6)  
in . Due to [3, equ. (4.3)],
(5.7) 
for a constant only depending on and , so the terms in the second, fourth and fifth line of (5.6) can be absorbed by the terms in the third line of (5.6). Then the righthand side of (5.6) is negative, a contradiction. Note, to estimate the term in the second line of (5.6) we used the homogeneity of ,
(5.8) 
and
(5.9) 
in where are positive constants depending only on . For further details we refer to [3, equ. (4.2), (4.3)].
(ii) We assume that . The quantity
(5.10) 
where a small and a large will be specified later, satisfies the evolution equation
(5.11)  
Assuming that is small we have in view of (5.1). We want to show that for all . For this we assume that there is a minimal and so that . Due to minimality of we conclude for all . Especially,
(5.12) 
and is strictly horospherically convex for all due to [3, Lemma 2.2]. We deduce that
(5.13) 
There holds
(5.14) 
with
(5.15) 
Note, that . From [3, Lemma 2.3] we deduce that
(5.16) 
in if . If then is umbilical point of , so let us write . Then we have
(5.17) 
and
(5.18) 
in . Using (5.11), the maximum principle, (5.4) and the fact that we conclude that
(5.19)  
in where
(5.20) 
For sufficiently small, the term
(5.21) 
is small and the terms in the second, fourth and fifth line of (5.19) can be absorbed by the terms in the third line of (5.19). Then the righthand side of (5.19) is negative if is large, a contradiction. ∎
6. The Euclidean case
The aim of this section is to prove Theorem 1.3 for the case . Due to the pinching estimates of the previous section we are in the situation that the proof in [17] basically carries over literally. For convenience of the reader, and since several elements of the proof simplify due to our pinching estimates, we give an outline of the arguments involved. Throughout this section it is understood that the assumptions of Theorem 1.3 hold.
We recall some simple observations. If the initial hypersurface is a sphere of radius , i.e. , then the flow hypersurfaces of the flow (1.11) remain spheres and for their radii at time we obtain the ODE
(6.1) 
with solution
(6.2) 
on where
(6.3) 
Hence the spherical flow blows up at time . The existence of a smooth solution to (1.11) up to a maximal time is well known, cf. [15, Sec. 2.5, Sec. 2.6].
From the avoidance principle we conclude the following corollary.
6.1 Corollary.
The next aim is to show that blows up, when the time approaches
6.2 Lemma.
The flow (1.11) only exists in a finite time interval and there holds
(6.6) 
Proof.
In view of Lemma 5.1 the hypersurfaces are convex and from (6.5) we deduce that the maximal time has to be finite. Due to the convexity we may write the flow hypersurfaces as radial graphs over the sphere,
(6.7) 
for some Then satisfies the scalar flow equation
(6.8) 
where the dot indicates the total time derivative, or
(6.9) 
when we consider the partial time derivative, cf. [15, p. 9899]. Under the assumption that is bounded, which is equivalent to we also obtain the estimate
(6.10) 
due to the convexity of and [15, Thm. 2.7.10]. To obtain a estimate, we need some curvature estimates. The proof is similar to the one for [17, Lemma 3.10, Lemma. 4.4]: Define the auxiliary function
(6.11) 
which, due to Lemma 3.1, satisfies the evolution equation
(6.12)  
At spatial maxima of there holds (recall )
(6.13)  
where we used
(6.14) 