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.
Acknowledgements. We would like to thank Professor Frank Morgan for pointing out the reference [H] to us, and Professor Mu-Tao Wang for useful comments and discussion.
2. Weighted isoperimetric inequality on multiply warped product manifolds
Let () be a product manifold. Equip with the warped product Riemannian metric for some , where is a Riemannian metric on the -dimensional manifold .
Denote the -dimensional volume of with respect to the product metric by , where . Define . We define the functions , and by
Here denotes that -dimensional volume of with respect to . Note that is convex if and only if is convex. For a bounded domain in , we define to be the region which has the same volume as , i.e. .
In this post, we will assume that and can possibly be zero only when . However, if is identified as a point, we do not assume is smooth at this point, e.g. metric with a conical singularity at this point is allowed. All functions considered in this post are assumed to be continuous.
One of the ingredients of the proof of Theorem 8 is the Jensen’s inequality. We notice that the Jensen’s inequality has also been used in [BBMP] to prove an isoperimetric result in . See also [H][Theorem 2.8, 2.9] for some related results.
Let us give some notations. Suppose is a monotone function on and is a probability measure on (i.e. ), we then define for a function
The following form of Jensen’s inequality will be useful to us.
Proposition 9 Let be a probability measure on and be functions on . Assume exists and is convex, then
Moreover, if is strictly increasing, then .
Proof: Define . Since is convex, by Jensen’s inequality,
If is strictly increasing, applying to the above inequality, we get .
We now prove Theorem 8. We remark that the reader may feel free to assume without affecting one’s understanding of the proof.
Proof: Assume first is piecewise , and that is a union of graphs over finitely many domains in . This means that there exists open, pairwise disjoint subsets of with Lipschitz boundary such that is represented by
By direct computation, the area element is given by
Here is the connection with respect to and is the -dimensional volume form on . Let
where we define , noting that .
On the other hand, for , we have
Comparing with (4), we have , and as is non-decreasing,
Finally, by the classical isoperimetric inequality,
In general, for a domain with Lipschitz boundary, we can approximate by piecewise domains which satisfy the above conditions. A standard approximation argument will then give the desired result.
Suppose the equality holds, we have two cases: (i) and (ii) .
Case (i). Let such that and . Then is clearly closed in . It is also open in because by (3), is locally constant on . Therefore and so is the slice .
Case (ii). In this case, we can without loss of generality assume on . The equality asserts that is a smooth hypersurface which has minimum area among all graphical hypersurfaces bounding the same volume, and so by the first variation formulas (e.g. [O][p. 1186]), if is a variation of with normal variation , then
for all such that
This implies that has constant mean curvature. It follows from [M][Corollary 7] that either has constant curvature and is a geodesic hypersphere, or is a slice .
- The monotonicity of is not assumed in Theorem 10 because of the same reason as Theorem 12. Also, from the proof of Theorem 8, we only need the classical isoperimetric inequality to hold for star-shaped domains for Theorem 10 to hold.
- By direction computation,
This condition is often easier to check as is usually not very explicit.
3. Isoperimetric inequalities involving weighted volume
In this section, we consider a variant of Theorem 8 involving a weighted volume. In particular, we prove an isoperimetric result without assuming the classical isoperimetric inequality to hold on .
where is the -dimensional volume form with respect to and is a radially symmetric weight function. Obviously, this is just the ordinary volume if . We define to be the region which has the same weighted volume as , i.e. . Our goal is to look for conditions such that
We define the functions and by
Proof: We use the same notations as the proof of Theorem 8. Since the proof is similar, we only indicate where changes are made.
We only prove the case where is piecewise , and that is a union of graphs over finitely many domains on . So we define as before. Note that by Condition (11). Let
As in (3),
where is defined by .
On the other hand, for , we have
As in (4), define by
As is convex, it is clear that we can proceed as in (7) to show that .
The analysis of the equality case and the general case is proved similarly as in Theorem 8.
Theorem 1 immediately follows from Theorem 11. For Theorem 2, note that the only place where we have used the monotonicity of (or in the context of Theorem 2) is (10). But since is star-shaped, and the monotonicity condition is not needed.
Let us state the following version for later use.
Theorem 12 Let be a bounded open set in with Lipschitz star-shaped boundary. Suppose is positive such that the function is convex, where . Then
The equality holds if and only if , i.e. is a coordinate slice.
Clearly, the classical isoperimetric inequality can also be expressed in the dual form: among all closed hypersurfaces in with the same area, the sphere encloses the largest volume. We now give the dual form of the weighted isoperimetric inequality.
Let be a bounded open set with Lipschitz boundary. We define to be such that
i.e. they have the same weighted boundary area.
Corollary 13 Let be a bounded open set in with Lipschitz boundary. Assume that
- The classical isoperimetric inequality holds: .
- is a non-negative function such that is strictly increasing and is non-negative and non-decreasing, where .
- The function is convex, where .
Proof: Let and . Then by Theorem 8 (with replaced by ),
So we have . Therefore, as is clearly increasing,
4. Some concrete examples
In this section, we provide some concrete examples of how Theorem 3 can be used to obtain some interesting geometric inequalities.
In all the examples below, the metric on is all given by , and the convexity of is directly checked by using (8). The computations have all been verified by Mathematica.
- On the Euclidean space , the warping function is . Choosing , we have
if . When , this is just the classical isoperimetric inequality.
- On the hyperbolic space , the warping function is . Choosing , we have
if . Similarly we also have
- On the open hemisphere , the warping function is , . Choosing , we have
if and . Similarly we also have
In all the above examples, we can convert the inequalities into a form which involves the volume of :
where . For example, in , the inequality
for , where is volume of the unit ball in . For other spaces, the inequality is not as explicit because is not explicit except when .
5. Weighted isoperimetric theorems involving higher order mean curvatures
In this section, we generalize the weighted isoperimetric inequality in a warped product manifold to some Minkowski-type inequalities involving the weighted integrals of the higher order mean curvatures. This is closely related to the quermassintegral inequalities ([GL]), which include as a special case the isoperimetric inequality, since the area integral can be interpreted as the integral of the zeroth mean curvature .
From now on, our ambient space is the warped product manifold equipped with the metric .
Before stating the main theorem, we give some definitions which are useful in studying the extrinsic geometry of hypersurfaces. On a hypersurface in , we define the normalized -th mean curvature function
where are the principal curvature functions on and the homogenous polynomial of degree is the -th elementary symmetric function
We adopt the usual convention .
The -th Newton transformation (cf. [R]) is useful in studying the extrinsic geometry of , and is defined as follows. If we write
then are given by
where is the second fundamental form of . One also defines , the identity map.
We define the vector field and the potential function
The warped product manifold is somewhat special in that there exists a nontrivial conformal Killing vector field, which in turn leads to some nice formulas of Hsiung-Minkowski types. We will need the following weighted Hsiung-Minkowski formulas (cf. [K1][Proposition 2.1], [KLP][Proposition 1]):
Proposition 14 (Weighted Hsiung-Minkowski formulas) Suppose is a smooth function on a closed hypersurface in , then for , we have
where is the unit normal vector, and is the tangential component of onto .
Proof: For completeness we sketch the proof here. Let . We compute
where is the inclusion of in and we used the fact that and . Applying the divergence theorem will then give the result.
Lemma 15 Suppose has constant curvature and is a star-shaped hypersurface with . Assume that for and . Then
This is essentially [KLP][Lemma 1, Proposition 1] or [BE][Section 2], despite some minor differences in the assumptions. (15) is proved in [KLP][Lemma 1 (2b)]. (15) follows the same proof as in [KLP][Proposition 1 (2)]. In the proof of [KLP][Proposition 1 (2)], it is assumed that (Condition (H4) in [Br], [BE]) and , but since we only require non-strict inequality in (12), the conclusion still holds under our assumption.
A remark is that we need to have constant curvature because conformal flatness of is essential in the formula of on p. 393 in [BE].
Theorem 16 Suppose is a domain in and its boundary is a smooth star-shaped hypersurface with . Assume that for and . Then for ,
The equality holds if and only if is a slice.
Proof: We will prove the following chain of inequalities:
Define . As , and , we have
which is clearly convex as . So by Theorem 12, we have
We now simply denote by and by . Applying the weighted Hsiung-Minkowski formula (Proposition 14), we have
By Theorem 12, the equality holds if and only if is a slice.
To relate the weighted volume to the integral of the higher order mean curvatures, we need stronger assumptions.
Theorem 17 Suppose has constant curvature and is a closed star-shaped hypersurface which is the boundary of a domain . Assume that for , and on . Then for ,
The equality holds if and only if is a slice.
The first two inequalities have already been proved in Theorem 16.
By Theorem 16, the equality holds if and only if is a slice.
- The Euclidean space with metric .
- The hyperbolic space with metric .
- The open hemisphere with metric .
In the following corollaries, we denote the point by and the volume of the unit ball in by . We obtain the following corollaries.
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 ,
Proof: The case where follows directly from Theorem 17. If , this is the ordinary isoperimetric inequality. So (13) is still true when , and we can perform induction starting from this case to show the assertion when .
Remark 2 If , the inequality becomes
In particular, if , it is easily seen from the proof that the assumption can be weakened to because is always positive. Corollary 18 extends (11) in Section 4 and also generalizes [KM] Theorem 2. See also [KM2][Theorem 2].
Corollary 19 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 ,
Corollary 20 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 ,
It is also possible to prove results analogous to Theorem 17 for standard space forms by extending the inequalities in Section 4 using the weighted Hsiung Minkowski inequalities, we will not do it here for the sake of simplicity.
6. Applications to eigenvalue estimates
First, we give an upper bound for the first eigenvalue of a differential operator related to the Newton’s tensor. We define to be the first eigenvalue of the symmetric second order differential operator on . The equality holds if and only if is immersed as a geodesic sphere. Note that is just the first Laplacian eigenvalue.
We now give an application of our main result to eigenvalues estimatation. The following theorem generalizes [WX] Theorem 1.2, which corresponds to the case where .
Theorem 21 Let be a closed embedded hypersurface in enclosing a region . Then
where and is the volume of the unit ball in . The equality holds if and only if is a hypersphere. (Note that is just the first Laplacian eigenvalue.)
Proof: By a suitable translation, we can assume that for .
By Theorem 3, we have
By the variational characterization of and the fact that , we have
Therefore combining the two inequalities we have
If the equality holds, then by Theorem 3, is a hypersphere.
Corollary 22 Let be a closed embedded hypersurface in enclosing a region . Then the first Laplacian eigenvalue on satisfies
The equality is attained if and only if is a ball.
To state our next result, we need to define the Steklov eigenvalues, as follows. Let be a compact Riemannian manifold with smooth boundary . The first nonzero Steklov eigenvalue is defined as the smallest of the following Steklov problem ([St])
where is the unit outward normal of . Physically, this describes the stationary heat distribution in a body whose flux through is proportional to the temperature on . It is known that the Steklov boundary problem (16) has a discrete spectrum
Moreover, has the following variational characterization (e.g. [KS] [Equation 2.3])
We will now prove an upper bound of with the techniques similar to that in Theorem 21.
Theorem 23 Suppose is a domain in the hyperbolic space with smooth boundary, then the first Steklov eigenvalue satisfies
after possibly a translation of the origin.
Proof: We use the hyperboloid model for :
with metric induced from the Minkowski metric . Intrisically, . By applying a rigid motion of , we can assume that for ,
From this we obtain
Remark 3 Theorem 23 is the hyperbolic analogue of the corresponding result for : , which is implied by [Brock] [Theorem 3].
We now prove a Pólya-Szegö inequality. For a function on , the Schwarz symmetrization ([S]) of is the non-increasing radial function such that for all ,
The classical Pólya-Szegö inequality then states that
if , where .
Theorem 24 With the same assumptions as Theorem 3, and assume is a convex non-decreasing function on . Then for any Lipschitz function on , we have
whenever the integral on the left is defined.
Note that if , and so by Jensen’s inequality,
The case where is similar. Letting , we get (18). On the other hand, by the coarea formula,
By definition, and so by the weighted isoperimetric inequality (Theorem 3),
So in view of (19), we have
where , noting that . Combining this with (18), we have
Integrating the above on and using a change of variable , we obtain
As an application of Theorem 24, we prove a Faber-Krahn inequality for the -Laplacian. Let be a bounded region with smooth boundary sitting inside the warped product manifold , where is the standard round metric. We assume and and regard as a point.
which is understood to hold in the weak sense. This is the Euler-Lagrange equation of the -Dirichlet integral
The study of the -Laplacian is of interest in the theory of non-Newtonian fluids (cf. [L]). The smallest eigenvalue is the smallest positive such that (20) holds for some non-trivial . It is known that
Theorem 25 Let , equipped with the metric . Suppose the classical inequality holds on and that , . Then for a domain in with smooth boundary, we have
Proof: Our assumptions ensure that is sufficiently smooth at . Suppose is the first eigenfunction on , which we can assume to be non-negative, and let be its Schwarz symmetrization. Then by Fubini’s theorem,
By Theorem 24, we have
Lastly, we prove a Cheeger type result for the weighted Laplacian eigenvalue problem and provide a lower bound for the weighted Cheeger’s constant.
Suppose is a Riemannian manifold with smooth non-empty boundary. Consider the weighted Laplacian eigenvalue problem in divergence form:
We define the weighted Cheeger’s constant to be
Theorem 26 For a compact domain with boundary,
Proof: We suspect this result is known to experts. We include the proof here since we cannot find an explicit reference. First, suppose is a non-negative function on with , then by the coarea formula and Fubini’s theorem,
Let be the first eigenfunction, which we can assume to be non-negative. Take in (25), then
From this and (24), we get .
It is not always easy to compute the (weighted) Cheeger’s constant. However, in the warped product setting and when is a radial function (but without assuming is symmetric), it can be readily estimated by the following result.
Theorem 27 Let , equipped with the metric with and . Suppose and satisfies the assumptions of Theorem 8, and assume is a bounded domain in with smooth boundary. Then
where . In particular,
Proof: By Theorem 8, we have
We conclude that
7. The necessity of the conditions
In this section, we examine the necessity of the conditions in Theorem 1, the classical isoperimetric inequality.
First, we consider the condition that being a convex function (Assumption 1), or equivalently that if is twice differentiable. We will show that this condition is necessary in the following sense:
Proposition 28 Suppose , then there exists a compact Riemannian manifold such that for the warped product manifold , the coordinate slice fails to be area-minimizing among nearby hypersurfaces enclosing the same volume.
Proof: The second variation formula for area on a constant-mean-curvature hypersurface , which is a critical point of the area functional subject to the constraint that a fixed amount of volume is enclosed, reads (e.g. [CY] [Equation (1)])
where the 1-parameter family of deformations of is given by with , is the square norm of the second fundamental form, is a function on and the normal variation is . In order to preserve the volume, we require .
Suppose . Choose a compact Riemannian manifold such that its first Laplacian eigenvalue and let be the corresponding eigenfunction. Such can, for example, be a sphere with sufficiently large radius such that .
Let equipped with the metric , then its straightforward to compute that
on the hypersurface . It is also easy to see that is an eigenfunction on with eigenvalue , so by the variational characterization of , satisfies and
Therefore the second variation formula becomes
It follows that fails to be area-minimizing among nearby hypersurfaces enclosing the same volume.
Remark 4 The necessity of the condition when is the unit sphere is also discussed in [LW]. See also [GLW] Remark 6.3. We notice that one of the conditions of Theorem 1.2 (isoperimetric inequality) in [GLW] is that . Proposition 28 does not contradict the result in [GLW] because it is assumed that in [GLW] while in our example, cannot hold. Indeed, as already noted in [GLW], the condition that guarantees that by Lichnerowicz’s theorem [Li].
It is also easy to see that the condition that the surjectivity (Assumption 1) of the projection map is necessary if , or equivalently, . Indeed, if , but as . If we take to be a small enough geodesic ball around a point which is far from , then the isoperimetric inequality clearly fails as has smaller area than , where .
Finally, we give an example in which the isoperimetric inequality fails when all assumptions except Assumption (1) hold. In view of Theorem 2, the counterexample must not be a star-shaped hypersurface. For convenience, we take the interval to be and to be any compact -dimensional manifold. Let which is a decreasing function. As is convex, (1) is satisfied. For the region , and . If we take and let , then we have but . Therefore the isoperimetric inequality fails.