The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality holds if and only if the curve is a circle. The classical isoperimetric inequality has been generalized to hypersurfaces in higher dimensional Euclidean space, and to various ambient spaces. For these generalizations, we refer to the beautiful article by Osserman [O] and the references therein. For a more modern account see [Ros]. Apart from two-dimensional manifolds and the standard space forms , and , there are few manifolds for which the isoperimetric surfaces are known. According to [BM], known examples include , , , , , , , and the Schwarzschild manifold, most of which are warped product manifolds over an interval or a circle. There are also many applications of the isoperimetric inequalities. For example, isoperimetric surfaces were used to prove the Penrose inequality [B], an inequality concerning the mass of black holes in general relativity, in some important cases.
In this post, we prove both classical and weighted isoperimetric results in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Some applications to geometric inequalities and eigenvalues are also given.
For the sake of simplicity, let us describe our main results on a warped product manifold. The multiply warped product case is only notationally more complicated and presents no additional conceptual difficulty.
Let () be a product manifold. Equip with the warped product Riemannian metric for some continuous , where is a Riemannian metric on the -dimensional manifold , which we assume to be compact and oriented. Define . We define the functions and by
Up to multiplicative constants, they are just the area of and the volume of respectively. For a bounded domain in , we define to be the region which has the same volume as , i.e. . We denote the area of and by and respectively.
One of our main results is the following isoperimetric theorem, which is a special case of Theorem 11.
We remark that our notion of convexity does not require the function to be differentiable: is convex on if and only if for any and .
One feature of our result is that except compactness, we do not impose any condition on the fiber manifold . We will see in Section 7 that without further restriction on or , our conditions are optimal in a certain sense.
The expression also has a number of geometric and physical meanings. It is related to the stability of the slice as a constant-mean-curvature (CMC) hypersurface (i.e. whether it is a minimizer of the area among nearby hypersurfaces enclosing the same volume). Indeed, it can be shown ([GLW] Proposition 6.2) that if and only if is a stable CMC hypersurface, where is the first Laplacian eigenvalue of . It is also related to the so called “photon spheres” in relativity (see [GLW] Proposition 6.1).
From (1), is convex if and only if is -convex. So if is non-decreasing and convex, then satisfies the convexity condition. One example of such a function is .
If is star-shaped in the sense that it is a graph over , i.e. of the form , we can remove the assumption on the monotonicity of .
Theorem 2 Suppose is a bounded open set in with Lipschitz boundary. Assume that
- is star-shaped.
- is convex, or equivalently, if is twice differentiable.
Then the isoperimetric inequality holds:
The equality holds if and only if is a coordinate slice .
If the classical isoperimetric inequality already holds on , we can extend it by the following result, which is a special case of Theorem 8.
Theorem 4 ([GLW][Theorem 1.2]) Suppose is a domain in with smooth star-shaped boundary. Assume that
- The Ricci curvature of satisfies , where is constant.
Then the isoperimetric inequality holds:
If , then the equality holds if and only if is a coordinate slice .
We note that our assumption in Theorem 2 complements that of [GLW]. This does not contradict the result in [GLW]. In fact, we will show the necessity of this and other conditions in Section 7 (cf. Proposition 28). We also notice that except the obvious case that has constant curvature, the equality holds only when is a coordinate slice. Indeed, combining Theorem 4 with Theorem 3, we can generalize Theorem 4 as follows.
Theorem 5 (Theorem 10) Suppose is a domain in with smooth star-shaped boundary. Assume that the Ricci curvature of satisfies and , where is constant. Suppose is a positive function such that is convex, where . Then the weighted isoperimetric inequality holds:
The equality holds if and only if either
- has constant curvature, is constant on , and is a geodesic hypersphere, or
- is a slice .
Theorem 6 Let be the product manifold equipped with the warped product metric .
- Suppose . Then the star-shaped isoperimetric hypersurfaces are precisely the coordinate slices .
- Suppose and where is constant. Then the star-shaped isoperimetric hypersurfaces are either geodesic hyperspheres if has constant curvature, or the coordinate slices .
- Suppose and where is constant. Then the coordinate slices cannot be isoperimetric hypersurfaces.
We also prove isoperimetric type theorems involving the integrals of higher order mean curvatures in warped product manifolds. For simplicity, let us state the result when the ambient space is (Corollary 18), which follows from a more general theorem (Theorem 17)
Theorem 7 (Corollary 18) Let be a closed embedded hypersurface in which is star-shaped with respect to and is the region enclosed by it. Assume that on . Then for any integer ,
where is the volume of the unit ball in . If , the equality holds if and only if is a hypersphere centered at .
Note that when , this reduces to , which is the classical isoperimetric inequality. In fact, we prove a stronger result (14) :
This can be compared to the following result of Guan-Li [GL][Theorem 2]:
under the same assumption.
The rest of this post is organized as follows. In Section 2, we first prove Theorem 3. In Section 3, we prove the isoperimetric inequality involving a weighted volume (Theorem 11), which implies Theorem 1 and Theorem 2. Although Theorem 3 can also be stated using the weighted volume, we prefer to prove the version involving only the ordinary volume for the sake of clarity, and indicates the changes needed to prove Theorem 11. In Section 4, we illustrate how we can obtain interesting geometric inequalities in space forms by using Theorem 8. In Section 5, we introduce the weighted Hsiung-Minkowski formulas in warped product manifolds, and combine them with the isoperimetric theorem to obtain new isoperimetric results involving the integrals of the higher order mean curvatures. In Section 6, we give further applications of our results to obtain some sharp eigenvalue estimates for some second order differential operators related to the extrinsic geometry of hypersurfaces and an eigenvalue estimate for the Steklov differential operator (also known as Dirichlet-to-Neumann map). A Pólya-Szegö inequality and a Faber-Krahn type theorem are also derived. Finally in Section 7, we show that the conditions of Theorem 1 are necessary by giving counterexamples where the isoperimetric inequality fails if any one of the conditions is violated.