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.

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 2Let 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

Therefore

Proposition 3Let be the family of smooth surfaces evolving according to (2), then

*Proof:* Let , then

We have

So

By proposition 2

Consider

So

As , we have

On the other hand, by [Ros] Theorem 1, we have

We deduce that

So

Proposition 4Let , then .

*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 **

*Proof:* The proof of Theorem 1 is now straightforward. Indeed the (1) holds if and only if , but this follows from the monotonicity property of by Proposition 3 and Proposition 4.

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.