Monotone quantities along hypersurfaces evolving under the inverse curvature flow have many applications in geometry and relativity. In [HI], Huisken and Ilmanen applied the monotone increasing property of Hawking mass to give a proof of the Riemannian Penrose Inequality. In a recent paper [BHW], Brendle, Hung and Wang discovered a monotone decreasing quantity along the inverse mean curvature flow in Anti-Desitter-Schwarzschild manifolds and used it to establish a Minkowski-type inequality for star-shaped hypersurfaces.
is monotone increasing along the (normalized) mean curvature flow (cf. also [LWW]).
Our motivation is as follows. In [K], the author used the Minkowski type integral formulas to obtain a series of inequalities relating the weighted -th mean curvature of a hypersurface. Indeed, it was shown in [K] that:
Theorem 2 Suppose is a closed hypersurface embedded in with for some . Then
Here and is the region enclosed by . The equality occurs if and only if is a sphere centered at .
On the other hand, in [KM], it was shown that a special case of the inequalities in (3) can be obtained by using the monotonic property of a related geometric quantity along the inverse mean curvature flow. The approach of using monotone quantities along various inverse curvature flow to derive geometric inequalities is a very fruitful and has attracted a lot of attention recently, see e.g. [BHW], [LG], and [LWW]. In view of the series of inequalities in Theorem 2, it is natural to ask: is there an associated series of geometric quantities which are monotone increasing along some inverse curvature flow in ? In this note, we give an affirmative answer to this question by showing that for each , there is a quantity which is monotone along the flow, as long as the initial hypersurface is star-shaped and -convex, i.e. .
Theorem 3 Let be a smooth, closed hypersurface embedded in with positive () and is star-shaped with respect to . Then
where is the distance to a fixed point and is the normalized -th mean curvature of . Furthermore, equality in (4) holds if and only if is a sphere centered at .
The rest of this note is organized as follows: in Section 1, we will collect all the necessary definitions and preliminary results, and we will prove our main theorem in Section 2. I would also like to thank Prof. Pengzi Miao for discussion.
Let us fix some notations. Let be a hypersurface and . Let be a local orthonormal frame on and let be the unit outward normal of . Let and be the connections on and respectively. We define the the shape operator by and is defined by . By abusing of notation, we will also denote the second fundamental form by . For simplicity, we will denote the dot product in by .
We define the -th mean curvature and the normalized -th mean curvature of by
respectively. Here is zero if or for some , or if as sets, otherwise it is defined as the sign of the permutation . For convenience, we also define and . The reason for this definition is that the Hsiung-Minkowski formula (Theorem 7) holds for (trivially).
Following [Reilly], we define the -th Newton transformation of as
For , is the identity. It is not hard to see that if are the principal directions, then
where are the principal curvatures.
From now on, we use the convention that repeated indices are summed from to . We collect some basic properties of and (e.g. [BC] Lemma 2.1):
where is a function on which is independent of . We have
Lemma 5 Under the flow in (5),
Proof: These equations are standard, and we only prove some of them for illustration. We have
The formula follows by the first variation formula for area. Finally, by Lemma 4,
Here we have used the fact that
Proposition 6 Under the flow in (5), for any function defined locally around ,
Proof: By Lemma 5,
The inverse curvature flow described in Theorem 1 actually exists for all time, given that the initial hypersurface is star-shaped:
Theorem 6 ([Gerhardt] and [Urbas]) } Let be a smooth, closed hypersurface in with positive , given by a smooth embedding . Suppose is star-shaped with respect to a point . Then the initial value problem
has a unique smooth solution , where is the unit outer normal vector to and is the normalized -th mean curvature of . Moreover, is star-shaped with respect to and the rescaled hypersurface , parametrized by , converges to a sphere centered at in the topology as .
Theorem 7 ([K] Cor. 3.1) Corollary 3.1} Suppose is a closed hypersurface in , and is a smooth function on . Then
Here is the position vector, is its tangential component and is the unit outward normal of .
2. Proof of the main results
For any function , we have
Now we fix , then and (the Euclidean metric). i.e.
This is equivalent to
where . Let , then the above equation becomes
On the other hand, by (7), we have
Combining this equation and (8), we have
In view of (9), this can also be written as
Suppose the equality holds, then by the equality case of the Newton’s inequality, must be umbilical, and therefore is a sphere.
Proof of Theorem 3: By Theorem 6, there exists a smooth solution to the inverse mean curvature flow with initial condition . Moreover, the rescaled hypersurface converges exponentially fast in the topology to a sphere. In particular, and hence , must be convex for large .
Let be a time when becomes convex. By using Theorem 7 twice, first with and then with , we have
Here we have used the fact that , which can be easily verified. As is convex, we have and , and hence (note that and are simultaneously diagonalizable), therefore
In other words, for defined in Theorem 1.
By Theorem 1, we know is monotone increasing, hence
which proves (4).
If the equality in (4) holds, then . It follows from the monotonicity of and the fact for large that
By Theorem 1, this implies that is a sphere for each . By the above argument, we have on , from which we can deduce that is a sphere centered at . Therefore, we conclude that the initial hypersurface is a sphere centered at .
Remark 1 We remark that the inequality in Theorem 3 also holds when by [KM] (using the convention that ). When , the assumption that is star-shaped can be dropped, by [K] Corollary 3.3.