In this note I am going to use inverse mean curvature flow to prove an inequality for star-shaped mean convex surface in the Euclidean space.
Theorem 1 Let be a closed embedded smooth hypersurface in which is star-shaped with respect to and has positive mean curvature , then
Here is the region enclosed by and is the distance from . The equality holds if and only if is a sphere.
This result generalizes a result in my previous post. I would like to thank Prof. Pengzi Miao for ideas.
0.1. Inverse mean curvature flow
In this section we will fix a closed embedded smooth hypersurface in given by the embedding which satisfies the assumptions in Theorem 1. The corresponding inverse mean curvature flow is then defined by the following equation:
where is the unit outward normal of . The hypersurface parametrized by will be denoted by and the region enclosed by will be denoted by .
By the results of [Gerhardt] and [Urbas], if the embedding of satisfies the assumptions in Theorem 1, then there exists a unique solution to (2) for all . Moreover, the rescaled hypersurfaces converge exponentially fast to a sphere in sense.
Recall that . The key idea to prove Theorem 1 is to observe that the following quantity is monotone:
Proposition 2 Let be the family of smooth surfaces evolving according to (2), then
Proof: We will use to denote the connection in . Let , then
Therefore . As is tangential, by using normal coordinates we have
Proposition 3 Let be the family of smooth surfaces evolving according to (2), then
Proof: Let , then
By proposition 2
As , we have
On the other hand, by [Ros] Theorem 1, we have
We deduce that
Proof: First of all, by Proposition 3, is increasing. In particular, exists (which can be ).
On the other hand, by the result of [Urbas], the rescaled surface given by converges exponentially fast in sense to a sphere. In particular, , and hence , must be convex (i.e. second fundamental form being positive-definite) for large enough . By Lemma 3.4 of this paper, for such , we have
We conclude that , but then by (3) again, we must have .
0.2. Proof of the main result
If the equality case holds, then clearly must be constant and so by Proposition 3 must be a sphere for all , in particular must be a sphere. The converse is easy to verify.