Weighted isoperimetric inequalities in warped product manifolds

1. Introduction

The classical isoperimetric inequality on the plane states that for a simple closed curve on {\mathbb R^2}, we have {L^2\ge 4\pi A}, where {L} is the length of the curve and {A} 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 {\mathbb R^n}, {\mathbb H^n} and {\mathbb S^n}, there are few manifolds for which the isoperimetric surfaces are known. According to [BM], known examples include {\mathbb R\times \mathbb H^n}, {\mathbb RP^3}, {\mathbb S^1\times \mathbb R^2}, {T^2\times \mathbb R}, {\mathbb R\times \mathbb S^n}, {\mathbb S^1\times \mathbb R^n}, {\mathbb S^1\times \mathbb S^2}, {\mathbb S^1\times \mathbb H^2} 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 {\displaystyle M=[0, l)\times N} ({l\le \infty}) be a product manifold. Equip {M} with the warped product Riemannian metric {\displaystyle g=dr^2+ s(r)^2 g_{N}} for some continuous {s(r)\ge 0}, where {g_{N}} is a Riemannian metric on the {m}-dimensional manifold {N}, which we assume to be compact and oriented. Define {B_R:=\{(r, \theta)\in M: r< R\}}. We define the functions {A(r)} and {v(r)} by

\displaystyle \begin{array}{rl} \displaystyle   A(r):= s(r)^m \; \textrm{ and }\; v(r):= \int_{0}^{r} A(t) dt. \end{array}

Up to multiplicative constants, they are just the area of {\partial B_r} and the volume of {B_r} respectively. For a bounded domain {\Omega} in {M}, we define {\Omega^\#} to be the region {B_R} which has the same volume as {\Omega}, i.e. {\mathrm{Vol} (B_R) =\mathrm{Vol} (\Omega) }. We denote the area of {\partial \Omega} and {\partial \Omega^\#} by {|\partial \Omega|} and {|\partial \Omega^\#|} respectively.

One of our main results is the following isoperimetric theorem, which is a special case of Theorem 11.

Theorem 1 Let {\Omega} be a bounded open set in {(M,g)} with Lipschitz boundary. Assume that

  1. The projection map {\pi: \partial \Omega\subset \mathbb [0, l)\times N\rightarrow N} defined by {(r, \theta)\mapsto \theta} is surjective.
  2. {s(r)} is non-decreasing.
  3. {A\circ v^{-1}} is convex, or equivalently, {s(r)s''(r)-s'(r)^2\ge 0} for {r>0} if {s} is twice differentiable.

Then the isoperimetric inequality holds:

\displaystyle \begin{array}{rl} \displaystyle |\partial \Omega|\ge|\partial \Omega^\#|.\end{array}

The equality holds if and only if {\partial \Omega} is a coordinate slice {\{r=\mathrm{constant}\}}.

We remark that our notion of convexity does not require the function to be differentiable: {f} is convex on {I} if and only if {f((1-t)x+ty)\le (1-t)f(x)+tf(y)} for any {t\in(0,1)} and {x, y\in I}.

One feature of our result is that except compactness, we do not impose any condition on the fiber manifold {N}. We will see in Section 7 that without further restriction on {N} or {s}, our conditions are optimal in a certain sense.

The expression {ss''-s'^2} comes from the observation that if {s} is twice differentiable, then as {v'=s^m}, the convexity of {A(v^{-1} (u))=s\left( v^{-1}\left( u \right)\right)^m} is equivalent to

\displaystyle \begin{array}{rl} \displaystyle  \frac{d^2}{du^2} A \left(v^{-1} (u)\right) =\frac{m}{ s(r)^{m+2}}\left(s(r)s''(r)-s'(r)^2\right)\ge 0, \ \ \ \ \ (1)\end{array}

where {r=v^{-1} (u)}.

The expression {ss''-s'^2} also has a number of geometric and physical meanings. It is related to the stability of the slice {\Sigma=\{r=r_0\}} 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 {\lambda_1(g_N)\ge m\left(s(r_0)'^2-s(r_0)s''(r_0)\right)} if and only if {\Sigma} is a stable CMC hypersurface, where {\lambda_1(g_N)} is the first Laplacian eigenvalue of {(N, g_N)}. It is also related to the so called “photon spheres” in relativity (see [GLW] Proposition 6.1).

From (1), {A\circ v^{-1}} is convex if and only if {s} is {\log}-convex. So if {f(r)} is non-decreasing and convex, then {s(r)=\exp(f(r))} satisfies the convexity condition. One example of such a function is {s(r)=e^r}.

If {\partial \Omega} is star-shaped in the sense that it is a graph over {N}, i.e. of the form {\partial \Omega=\{(r, \theta): r=\rho(\theta), \theta\in N\}}, we can remove the assumption on the monotonicity of {s(r)}.

Theorem 2 Suppose {\Omega} is a bounded open set in {(M,g)} with Lipschitz boundary. Assume that

  1. {\partial \Omega} is star-shaped.
  2. {A\circ v^{-1}} is convex, or equivalently, {ss''-s'^2\ge 0} if {s} is twice differentiable.

Then the isoperimetric inequality holds:

\displaystyle \begin{array}{rl} \displaystyle |\partial \Omega|\ge|\partial \Omega^\#|.\end{array}

The equality holds if and only if {\partial \Omega} is a coordinate slice {\{r=\mathrm{constant}\}}.

If the classical isoperimetric inequality already holds on {(M,g)}, we can extend it by the following result, which is a special case of Theorem 8.

Theorem 3 (Weighted isoperimetric inequality) Let {\Omega} be a bounded open set in {(M,g)} with Lipschitz boundary. Assume that

  1. The classical isoperimetric inequality holds on {(M,g)}, i.e. {|\partial \Omega|\ge|\partial \Omega^\#|}.
  2. {a(r)} is a non-negative function such that {\psi(r):=b(r)A(r)} is non-decreasing and non-negative, where {b(r):=a(r)-a(0)}.
  3. The function {\psi\circ v^{-1} } is convex.

Then

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} a(r)dS \ge\int_{\partial \Omega^{\#}} a(r)dS. \end{array}

If {b(r)A(r)>0} for {r>0}, then the equality holds if and only if {\partial \Omega} is a coordinate slice {\{r=\mathrm{constant}\}}.

Using a volume preserving flow, recently Guan, Li and Wang [GLW] (see also [GL2]) proved the following related result, assuming {s} is smooth ({\partial \Omega} is called “graphical” in [GLW]):

Theorem 4 ([GLW][Theorem 1.2]) Suppose {\Omega} is a domain in {(M,g)} with smooth star-shaped boundary. Assume that

  1. The Ricci curvature {\mathrm{Ric}_N} of {g_N} satisfies {\mathrm{Ric}_N\ge (m-1)K g_N}, where {K>0} is constant.
  2. {0\le s'^2-ss''\le K}.

Then the isoperimetric inequality holds:

\displaystyle \begin{array}{rl} \displaystyle |\partial \Omega|\ge|\partial \Omega^\#|.\end{array}

If { s'^2-ss''<K}, then the equality holds if and only if {\partial \Omega} is a coordinate slice {\{r=r_0\}}.

We note that our assumption {ss''-s'^2\ge 0} 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 {M} has constant curvature, the equality holds only when {\partial \Omega} is a coordinate slice. Indeed, combining Theorem 4 with Theorem 3, we can generalize Theorem 4 as follows.

Theorem 5 (Theorem 10) Suppose {\Omega} is a domain in {(M,g)} with smooth star-shaped boundary. Assume that the Ricci curvature {\mathrm{Ric}_N} of {g_N} satisfies {\mathrm{Ric}_N\ge (m-1)K g_N} and {0\le s'^2-ss''\le K}, where {K>0} is constant. Suppose {a(r)} is a positive function such that {b( v^{-1} (u))s( v^{-1} (u))^m} is convex, where {b(r):=a(r)-a(0)}. Then the weighted isoperimetric inequality holds:

\displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega}a(r)dS\ge \int_{\partial \Omega^\#}a(r)dS.\end{array}

The equality holds if and only if either

  1. {(M,g)} has constant curvature, {a(r)} is constant on {\partial \Omega}, and {\partial \Omega} is a geodesic hypersphere, or
  2. {\partial \Omega} is a slice {\{r=r_0\}}.

Combining Theorem 4, Theorem 2 and the proof of Proposition 28, we get the following picture for the isoperimetric problem in warped product manifolds:

Theorem 6 Let {M} be the product manifold {[0, l)\times N} equipped with the warped product metric {g=dr^2+s(r)^2 g_N}.

  1. Suppose {s'^2-ss''\le 0}. Then the star-shaped isoperimetric hypersurfaces are precisely the coordinate slices {\{r=r_0\}}.
  2. Suppose {0\le s'^2-ss''\le K} and {\mathrm{Ric}_N\ge (m-1)Kg_N} where {K>0} is constant. Then the star-shaped isoperimetric hypersurfaces are either geodesic hyperspheres if {(M,g)} has constant curvature, or the coordinate slices {\{r=r_0\}}.
  3. Suppose {s'^2-ss''>K} and {\lambda_1(g_N)\le mK} where {K>0} is constant. Then the coordinate slices {\{r=r_0\}} cannot be isoperimetric hypersurfaces.

We also prove isoperimetric type theorems involving the integrals of higher order mean curvatures {(H_k)} in warped product manifolds. For simplicity, let us state the result when the ambient space is {\mathbb R^n} (Corollary 18), which follows from a more general theorem (Theorem 17)

Theorem 7 (Corollary 18) Let {\Sigma} be a closed embedded hypersurface in {\mathbb R^{m+1}} which is star-shaped with respect to {0} and {\Omega} is the region enclosed by it. Assume that {H_k>0} on {\Sigma}. Then for any integer {l\ge 0},

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n ^{-\frac{l-1}{n}}\mathrm{Vol} (\Omega)^{\frac{n-1+l}{n}} \le\int_{\Sigma} H_k r^{l+k}dS, \end{array}

where {\beta_n} is the volume of the unit ball in {\mathbb R^n}. If {l\ge 1}, the equality holds if and only if {\Sigma} is a hypersphere centered at {0}.

Note that when {k=l=0}, this reduces to {n {\beta_n}^{\frac{1}{n}}\mathrm{Vol} (\Omega)^{\frac{n-1}{n}} \le \mathrm{Area} (\Sigma) }, which is the classical isoperimetric inequality. In fact, we prove a stronger result (14) :

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n^{-\frac{l-1}{n}}\mathrm{Vol} (\Omega)^{\frac{n-1+l}{n}} \le \int_{\partial \Omega} H_0 r ^{l} dS \le \int_{\partial \Omega} H_1 r ^{l+1} dS \le \cdots \le \int_{\partial \Omega} H_k r ^{l+k} dS. \end{array}

This can be compared to the following result of Guan-Li [GL][Theorem 2]:

\displaystyle \begin{array}{rl} \displaystyle   \left(\frac{1}{ \beta_n }\mathrm{Vol} (\Omega)\right)^{\frac{1}{n}}\le \left(\frac{1}{n\beta_n}{\int_\Sigma H_0dS}\right)^{\frac{1}{n-1}}\le \cdots \le \left(\frac{1}{n\beta_n} \int_\Sigma H_k dS \right) ^{\frac{1}{n-k-1}} \end{array}

under the same assumption.

Some applications of the weighted isoperimetric inequalities will also be given in Section 4 and Section 6.

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

In this section, we first prove Theorem 3. As explained in Section 1, our result actually applies to multiply warped product manifold with no additional difficulties. Let us now describe our setting.

Let {\displaystyle M=[0, l)\times \prod_{q=1}^{p}N_q} ({l\le \infty}) be a product manifold. Equip {M} with the warped product Riemannian metric {\displaystyle g:=dr^2+ \sum_{q=1}^{p}s_q(r)^2 g_{N_q}} for some {s_q(r)\ge 0}, where {g_{N_q}} is a Riemannian metric on the {m_q}-dimensional manifold {N_q}.

Denote the {m}-dimensional volume of {\displaystyle N:=\prod_{q=1}^{p}N_q} with respect to the product metric {\displaystyle \sum_{q=1}^{p}g_{N_q}} by {|N|} , where {\displaystyle m=\sum_{q=1}^{p}m_q}. Define {B_R:=\{(r, \theta)\in M: r< R\}}. We define the functions {A(r)}, {v(r)} and {V(r)} by

\displaystyle \begin{array}{rl} \displaystyle   A(r):= \prod_{q=1}^{p}s_q(r)^{m_q},\; v(r):= \int_{0}^{r} A(t) dt \;\textrm{ and }\; V(r):= |B_r| =|N| v(r). \end{array}

Here {|B_R|} denotes that {(m+1)}-dimensional volume of {B_R} with respect to {g}. Note that {\psi\circ v^{-1}} is convex if and only if {\psi\circ V^{-1}} is convex. For a bounded domain {\Omega} in {M}, we define {\Omega^\#} to be the region {B_R} which has the same volume as {\Omega}, i.e. {|B_R|=|\Omega|}.

In this post, we will assume that {A(r)>0} and can possibly be zero only when {r=0}. However, if {\{r=0\}} is identified as a point, we do not assume {g} 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.

Theorem 8 Let {\Omega} be a bounded open set in {(M,g)} with Lipschitz boundary. Assume that

  1. The classical isoperimetric inequality holds on {(M,g)}, i.e. {|\partial \Omega|\ge|\partial \Omega^\#|}.
  2. {a(r)} is a non-negative function such that {b(r)A(r)} is non-decreasing and non-negative, where {b(r):=a(r)-a(0)}.
  3. The function {b (V^{-1} (u)) \;A (V^{-1} (u))} is convex.

Then

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} a(r)dS \ge\int_{\partial \Omega^{\#}} a(r)dS. \end{array}

If {b(r)A(r)>0} for {r>0}, then the equality holds if and only if {r=\mathrm{constant}}, i.e. {\partial \Omega} is a coordinate slice.

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 {\mathbb R^n}. See also [H][Theorem 2.8, 2.9] for some related results.

Let us give some notations. Suppose {\phi} is a monotone function on {\mathbb R} and {\mu} is a probability measure on {X} (i.e. {\mu(X)=1}), we then define for a function {\rho: X\rightarrow \mathbb R}

\displaystyle \begin{array}{rl} \displaystyle \mathcal M_{\phi,\mu}[\rho]:=\phi^{-1}\left(\int_X \phi(\rho)d\mu\right).\end{array}

The following form of Jensen’s inequality will be useful to us.

Proposition 9 Let {\mu} be a probability measure on {X} and {\phi, \psi} be functions on {\mathbb R}. Assume {\phi^{-1}} exists and {\psi\circ \phi^{-1}} is convex, then

\displaystyle \begin{array}{rl} \displaystyle \psi \left(\mathcal M_{\phi, \mu}[\rho]\right) \le\int_{X} \psi (\rho) d\mu. \end{array}

Moreover, if {\psi} is strictly increasing, then {\mathcal M_{\phi, \mu}[\rho] \le \mathcal M_{\psi, \mu}[\rho]}.

Proof: Define {\Phi=\psi\circ \phi^{-1}}. Since { {\Phi}} is convex, by Jensen’s inequality,

\displaystyle \begin{array}{rl} \displaystyle   \psi \left(\mathcal M_{\phi, \mu}[\rho]\right) =\Phi\left(\int_X \phi (\rho) d\mu\right) \le \int_{X}\Phi (\phi (\rho)) d\mu = \int_{X} \psi (\rho) d\mu. \end{array}

If {\psi} is strictly increasing, applying {\psi^{-1}} to the above inequality, we get {\mathcal M_{\phi, \mu}[\rho] \le \mathcal M_{\psi, \mu}[ \rho]}. \Box

We now prove Theorem 8. We remark that the reader may feel free to assume {p=1} without affecting one’s understanding of the proof.
Proof: Assume first {\Sigma=\partial \Omega} is piecewise {C^1}, and that {\Sigma} is a union of graphs over finitely many domains in {N}. This means that there exists open, pairwise disjoint subsets {\{S_i\}_{i=1}^l} of {N} with Lipschitz boundary such that {\Sigma} is represented by

\displaystyle \begin{array}{rl} \displaystyle   \Sigma=\partial \Omega=\left\{(r, \theta): r=r_{i,j} (\theta), \theta\in \overline S_i, j\in\{1,\cdots, 2k_i\}, i\in\{1, \cdots, l\}\right\} \end{array}

and

\displaystyle \begin{array}{rl} \displaystyle   \overline \Omega =\left\{(r, \theta): r_{i, 2\kappa-1} (\theta)\le r\le r_{i, 2\kappa} (\theta), \theta\in \overline S_i, \kappa\in\{1,\cdots, k_i\}, i\in\{1,\cdots, l\}\right\}, \end{array}

where

\displaystyle \begin{array}{rl}  &r_{i,j}\in C^1(S_i)\cap C^0(\overline S_i),\quad j=1,\cdots, 2k_i, \\ &r_{i,1} (\theta)<\cdots< r_{i,2k_i} (\theta)\quad \textrm{for } \theta\in S_i, \end{array}

\displaystyle \begin{array}{rl}  &r_{i,1} (\theta) \begin{cases} =0\quad \textrm{if } (0, \theta)\in \Omega\\ >0\quad \textrm{if } (0, \theta)\notin \Omega. \end{cases}  \end{array}

By direct computation, the area element is given by

\displaystyle \begin{array}{rl} \displaystyle   dS =\prod_{q=1}^{p}\left(1+s_q(r)^{-2}|\nabla_{N_q} \, r|^2_{g_{N_q}}\right)^{\frac{1}{2}}s_q(r)^{m_q}d\mathrm{vol}_N. \end{array}

Here {\nabla_{N_q} } is the connection with respect to {g_{N_q}} and {d\mathrm{vol}_N} is the {m}-dimensional volume form on {N}. Let

\displaystyle \begin{array}{rl} \displaystyle   I:=\int_{\partial \Omega} b(r)dS\textrm{ and }I^{\#}:=\int_{\partial \Omega^{\#}} b(r)dS.  \end{array}

We claim that

\displaystyle \begin{array}{rl} \displaystyle  I\ge I^\#. \ \ \ \ \ (2)\end{array}

Let {\displaystyle S=\bigcup_{i=1}^l S_i} and {\psi(r)=b(r)A(r)=b(r)\prod_{q=1}^{p}s_q(r)^{m_q}}. First of all,

\displaystyle \begin{array}{rl} \displaystyle   I =& \displaystyle \sum_{i=1}^{l}\sum_{j=1}^{2k_i}\int_{S_i}b(r_{i,j})\prod_{q=1}^{p}\left(1+s_q(r_{i,j})^{-2}\left|\nabla_{N_q}\,r_{i,j}\right|^2_{g_{N_q}}\right)^{\frac{1}{2}}s_q(r_{i,j})^{m_q}d\mathrm{vol}_N\\ \ge& \displaystyle \sum_{i=1}^{l}\sum_{j=1}^{2k_i}\int_{S_i}b(r_{i,j})\prod_{q=1}^{p}s_q(r_{i,j})^{m_q}d\mathrm{vol}_N\\ =& \displaystyle \sum_{i=1}^{l}\sum_{j=1}^{2k_i}\int_{S_i} \psi(r_{i,j}) d\mathrm{vol}_N\\ \ge& \displaystyle \sum_{i=1}^{l}\int_{S_i} \psi(r_{i, 2k_i}) d\mathrm{vol}_N\\ =& \displaystyle \int_{N}\psi(\rho) d\mathrm{vol}_N  \ \ \ \ \ (3)\end{array}

where we define {\rho(\theta):= \begin{cases} r_{i,2k_i} (\theta)\; & \displaystyle \textrm{if }\theta\in S_i,\\ 0 \;& \displaystyle \textrm{if }\theta\in N\setminus S \end{cases} }, noting that {b(0)=0}.

On the other hand, for {B_R=\Omega^{\#}}, we have

\displaystyle \begin{array}{rl} \displaystyle   |B_R| = |N| \int_{0}^{R} A(r)dr=|N|v(R). \end{array}

As {|B_R|=|\Omega|}, it is not hard to see by Fubini’s theorem that

\displaystyle \begin{array}{rl} \displaystyle  |B_R| =|\Omega|=\sum_{i=1}^{l}\sum_{j=1}^{2k_i} (-1)^j\int_{S_i} v(r_{i,j}) d\mathrm{vol}_N. \ \ \ \ \ (4)\end{array}

Define {R_1} by

\displaystyle \begin{array}{rl} \displaystyle   \sum_{i=1}^{l}\sum_{j=1}^{2k_i} (-1)^j\int_{S_i} v(r_{i,j}) d\mathrm{vol}_N \le& \displaystyle \sum_{i=1}^{l}\int_{S_i} v(r_{i,2k_i}) d\mathrm{vol}_N=: |B_{R_1}|, \end{array}

noting that {v(r)} is increasing. Then by the definition of {V} and {\rho},

\displaystyle \begin{array}{rl} \displaystyle  R_1 = V^{-1} \left(\int_{N} v(\rho) d\mathrm{vol}_N\right). \ \ \ \ \ (5)\end{array}

Comparing with (4), we have {R_1\ge R}, and as {\psi(r)} is non-decreasing,

\displaystyle \begin{array}{rl} \displaystyle  I^{\#}=|N|b(R)A(R)=|N|\psi(R)\le |N| \psi(R_1). \ \ \ \ \ (6)\end{array}

Now take {d\mu=\frac{1}{|N|}d\mathrm{vol}_N}. As {\psi\circ V^{-1}} is convex, by (3), Jensen’s inequality (Proposition 9), (5) and (6),

\displaystyle \begin{array}{rl} \displaystyle   \frac{1}{|N|}I \ge\frac{1}{|N|}\int_{N}\psi(\rho) d\mathrm{vol}_N =& \displaystyle \int_{N}\psi(\rho) d\mu\\ \ge& \displaystyle  \psi \left(\mathcal M_{V, \mu}[\rho]\right)\\ =& \displaystyle  \psi \left(V^{-1}\left( \int_N V (\rho) d\mu\right)\right)\\ =& \displaystyle  \psi\left(V^{-1}\left(\int_{N} v(\rho)d\mathrm{vol}_N \right)\right)\\ =& \displaystyle \psi(R_1)\\ \ge& \displaystyle \frac{1}{|N|}I^\#.  \ \ \ \ \ (7)\end{array}

We have proved (2). Moreover, from (3), if {\psi(r)>0} for {r>0}, then {r_{i,j}} must be locally constant and hence {r} is constant on {\Sigma}.

Finally, by the classical isoperimetric inequality,

\displaystyle \begin{array}{rl} \displaystyle    \int_{\partial \Omega} a(r)dS =& \displaystyle I+ a(0)\int_{\partial \Omega} dS\\ \ge & \displaystyle I^{\#}+ a(0)\int_{\partial \Omega^{\#}} dS\\ =& \displaystyle \int_{\partial \Omega^{\#}} a(r)dS.   \end{array}

In general, for a domain {\Omega} with Lipschitz boundary, we can approximate {\Omega} by piecewise {C^1} domains {\Omega_i} which satisfy the above conditions. A standard approximation argument will then give the desired result. \Box

As explained in Section 1, combining Theorem 4 ([GLW][Theorem 1.2]) with Theorem 8 we have

Theorem 10 Suppose {\Omega} is a domain in {(M,g)} with smooth star-shaped boundary. Assume that the Ricci curvature {\mathrm{Ric}_N} of {g_N} satisfies {\mathrm{Ric}_N\ge (m-1)K g} and {0\le s'^2-ss''\le K}, where {K>0} is constant. Suppose {a(r)} is a positive function such that {b( v^{-1} (u))s( v^{-1} (u))^m} is convex, where {b(r):=a(r)-a(0)}. Then the weighted isoperimetric inequality holds:

\displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega}a(r)dS\ge \int_{\partial \Omega^\#}a(r)dS.\end{array}

The equality holds if and only if either

  1. {(M,g)} has constant curvature, {a(r)} is constant on {\partial \Omega} and {\partial \Omega} is a geodesic hypersphere, or
  2. {\partial \Omega} is a slice {\{r=r_0\}}.

Proof: The inequality follows from Theorem 4 ([GLW][Theorem 1.2]) and Theorem 8 (see also Remark 1 (1)).

Suppose the equality holds, we have two cases: (i) {a(r)|_{\partial \Omega}\not\equiv a(0)} and (ii) {a(r)|_{\partial \Omega}\equiv a(0)}.
Case (i). Let {p\in \partial \Omega} such that {a(r(p))\ne a(0)} and {r_0=r(p)}. Then {S=\{q\in \partial \Omega: r(q)=r_0\}} is clearly closed in {\partial \Omega}. It is also open in {\partial \Omega} because by (3), {r} is locally constant on {\{q\in \partial \Omega: a(r(q))\ne a(0)\}}. Therefore {S=\partial \Omega} and so {\partial \Omega} is the slice {\{r=r_0\}}.
Case (ii). In this case, we can without loss of generality assume {a\equiv 1} on {\partial \Omega}. The equality asserts that {\partial \Omega} 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 {\Sigma_t} is a variation of {\Sigma} with normal variation {u \nu_{\Sigma_t}}, then

\displaystyle \begin{array}{rl} \displaystyle   \left.\frac{d}{dt}\right|_{t=0}\mathrm{Area} (\Sigma_t) = m \int_{\Sigma} u H_1 dS=0 \end{array}

for all {u} such that

\displaystyle \begin{array}{rl} \displaystyle \left.\frac{d}{dt}\right|_{t=0}\mathrm{Vol} (\Omega_t) = \int_{\Sigma} u dS=0.\end{array}

This implies that {\Sigma} has constant mean curvature. It follows from [M][Corollary 7] that either {(M,g)} has constant curvature and {\partial \Omega} is a geodesic hypersphere, or {\partial \Omega} is a slice {\{r=r_0\}}. \Box

Remark 1

  1. The monotonicity of {s(r)} 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.
  2. By direction computation,

    \displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \frac{d^2}{du^2}\left(b( v^{-1} (u)) \;s (v^{-1} (u))^m\right)\\ =& \displaystyle \frac{1}{s(r)^{m+2}}\left(s(r)^2b''(r)+ms(r)s'(r)b'(r)+mb(r)\left(s(r)s''(r)-s'(r)^2\right)\right), \end{array}

    where {r=v^{-1} (u)}. So the convexity of {b( v^{-1} (u)) \;s( v^{-1} (u))^m} can be rephrased as

    \displaystyle \begin{array}{rl} \displaystyle  s(r)^2b''(r)+ms(r)s'(r)b'(r)-mb(r)\left(s'(r)^2-s(r)s''(r)\right)\ge 0. \ \ \ \ \ (8)\end{array}

    This condition is often easier to check as {v^{-1}} 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 {M}.

We consider the weighted volume defined by

\displaystyle \begin{array}{rl} \displaystyle  \mathrm{Vol}_c(\Omega):=\int_{\Omega}c(r)dv_g \ \ \ \ \ (9)\end{array}

where {dv_g} is the {(m+1)}-dimensional volume form with respect to {g} and {c(r)> 0} is a radially symmetric weight function. Obviously, this is just the ordinary volume if {c\equiv 1}. We define {\widetilde \Omega^\#} to be the region {B_R} which has the same weighted volume as {\Omega}, i.e. {\mathrm{Vol}_c(\widetilde \Omega^\#)=\mathrm{Vol}_c(\Omega)}. Our goal is to look for conditions such that

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega}a(r)dS \ge\int_{\partial \widetilde \Omega^\#}a(r)dS. \end{array}

We define the functions {\widetilde v(r)} and {\widetilde V(r)} by

\displaystyle \begin{array}{rl} \displaystyle   \widetilde v(r):= \int_{0}^{r}c(t) A(t) dt\; \textrm{ and }\; \widetilde V(r):= \int_{B_r}c \,dv_g =|N|\widetilde v(r). \end{array}

Theorem 11 Let {\Omega} be a bounded open set in {(M,g)} with Lipschitz boundary. Assume that

  1. The projection map {\pi: \partial \Omega\rightarrow N} defined by {(r, \theta)\mapsto \theta} is surjective.
  2. {a(r)} is a positive function such that {\psi(r):=a(r)A(r)} is non-decreasing,
  3. The function {\psi\circ \widetilde V^{-1}} is convex.

Then

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} a(r)dS \ge\int_{\partial \widetilde\Omega^{\#}} a(r)dS. \end{array}

The equality holds if and only if {r=\mathrm{constant}}, i.e. {\partial \Omega} is a coordinate slice.

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 {\Sigma=\partial \Omega} is piecewise {C^1}, and that {\Sigma} is a union of graphs over finitely many domains on {N}. So we define {r_{i,j}} as before. Note that {N=\bigcup_{i=1}^l \overline {S_i}} by Condition (11). Let

\displaystyle \begin{array}{rl} \displaystyle   I:=\int_{\partial \Omega} a(r)dS\textrm{ and }I^{\#}:=\int_{\partial \widetilde\Omega^{\#}} a(r)dS.  \end{array}

As in (3),

\displaystyle \begin{array}{rl} \displaystyle    I =& \displaystyle \sum_{i=1}^{l}\sum_{j=1}^{2k_i}\int_{S_i}a(r_{i,j})\prod_{q=1}^{p}\left(1+s_q(r_{i,j})^{-2}\left|\nabla_{N_q}\, r_{i,j}\right|^2_{g_{N_q}}\right)^{\frac{1}{2}}s_q(r_{i,j})^{m_q}d\mathrm{vol}_N\\ \ge& \displaystyle \sum_{i=1}^{l}\int_{S_i} \psi(r_{i, 2k_i}) d\mathrm{vol}_N\\ =& \displaystyle \int_{N}\psi(\rho) d\mathrm{vol}_N   \end{array}

where {\rho: N\rightarrow [0, \infty)} is defined by {\rho(\theta):= \max\{r: (r, \theta)\in \Sigma\} }.

On the other hand, for {B_R=\widetilde\Omega^{\#}}, we have

\displaystyle \begin{array}{rl} \displaystyle   \mathrm{Vol}_c (B_R) = |N| \int_{0}^{R} c(r)A(r)dr=|N|\widetilde v(R). \end{array}

As in (4), define {R_1} by

\displaystyle \begin{array}{rl} \displaystyle   \mathrm{Vol}_c (B_R) = \mathrm{Vol}_c(\Omega ) =& \displaystyle \sum_{i=1}^{l}\sum_{j=1}^{2k_i} (-1)^j\int_{S_i} \widetilde v(r_{i,j}) d\mathrm{vol}_N\\ \le& \displaystyle \sum_{i=1}^{l}\int_{S_i} \widetilde v(r_{i,2k_i}) d\mathrm{vol}_N=: \mathrm{Vol}_c(B_{R_1}). \end{array}

Then analogous to (5) and (6), we have

\displaystyle \begin{array}{rl} \displaystyle  R_1 = \widetilde V^{-1} \left(\int_{N} \widetilde v(\rho) d\mathrm{vol}_N\right) \; \textrm{and}\; I^{\#}=|N|\psi(R)\le |N| \psi(R_1). \ \ \ \ \ (10)\end{array}

As {\psi\circ \widetilde V^{-1}} is convex, it is clear that we can proceed as in (7) to show that {I\ge I^\#}.

The analysis of the equality case and the general case is proved similarly as in Theorem 8. \Box

Theorem 1 immediately follows from Theorem 11. For Theorem 2, note that the only place where we have used the monotonicity of {\psi} (or {s} in the context of Theorem 2) is (10). But since {\Sigma} is star-shaped, {R_1=R} and the monotonicity condition is not needed.

Let us state the following version for later use.

Theorem 12 Let {\Omega} be a bounded open set in {(M,g)} with Lipschitz star-shaped boundary. Suppose {a(r)} is positive such that the function {\psi\circ \widetilde V^{-1}} is convex, where {\psi(r)=a(r)A(r)}. Then

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} a(r)dS \ge\int_{\partial \widetilde\Omega^{\#}} a(r)dS. \end{array}

The equality holds if and only if {r=\mathrm{constant}}, i.e. {\partial \Omega} is a coordinate slice.

Clearly, the classical isoperimetric inequality can also be expressed in the dual form: among all closed hypersurfaces in {\mathbb R^n} with the same area, the sphere encloses the largest volume. We now give the dual form of the weighted isoperimetric inequality.

Let {\Omega} be a bounded open set with Lipschitz boundary. We define {\widetilde\Omega^*} to be {B_{R^*}} such that

\displaystyle \begin{array}{rl} \displaystyle \int_{\partial B_{R^*}}a(r)dS=\int_{\partial \Omega}a(r)dS,\end{array}

i.e. they have the same weighted boundary area.

It should now be clear that Theorem 8 is also true if we replace {V} by {\widetilde V}. Although rather obvious, we would like to record the dual version of Theorem 8 here.

Corollary 13 Let {\Omega} be a bounded open set in {(M,g)} with Lipschitz boundary. Assume that

  1. The classical isoperimetric inequality holds: {| \partial \Omega|\ge| \partial \Omega^\#|}.
  2. {a(r)} is a non-negative function such that {a(r)A(r)} is strictly increasing and {b(r)A(r)} is non-negative and non-decreasing, where {b(r)=a(r)-a(0)}.
  3. The function {\psi\circ \widetilde V^{-1} } is convex, where {\psi(r)=b(r)A(r)}.

Then

\displaystyle \begin{array}{rl} \displaystyle   \int_\Omega c(r)dv_g \le\int_{\widetilde\Omega^*} c(r)dv_g. \end{array}

Proof: Let {\widetilde \Omega^\#=B_{R^{\#}}} and {\widetilde \Omega^*=B_{R^{*}}}. Then by Theorem 8 (with {V} replaced by {\widetilde{V}}),

\displaystyle \begin{array}{rl} \displaystyle   |N|a(R^{*})A(R^{*}) = & \displaystyle \int_{\partial B_{R^*}}a(r)dS\\ =& \displaystyle \int_{\partial \Omega}a(r)dS\\ \ge& \displaystyle  \int_{\partial \widetilde\Omega^\#}a(r)dS\\ =& \displaystyle |N|a(R^{\#})A(R^{\#}). \end{array}

So we have {R^*\ge R^{\#}}. Therefore, as {\widetilde V} is clearly increasing,

\displaystyle \begin{array}{rl} \displaystyle   \int_{\widetilde\Omega^*} c(r)dv_g =\mathrm{Vol}_c(\widetilde\Omega^*) =\widetilde V(R^*) \ge\widetilde V(R^\#) =\mathrm{Vol}_c(\widetilde\Omega^\#) =& \displaystyle \int_\Omega c(r)dv. \end{array}

\Box

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 {g} on {M} is all given by {g=dr^2+s(r)^2 g_{\mathbb S^{m}}}, and the convexity of {b(V^{-1} (u)) s(V^{-1} (u))^m} is directly checked by using (8). The computations have all been verified by Mathematica.

  1. On the Euclidean space {\mathbb R^{n}}, the warping function is {s(r)= r}. Choosing {a(r)=b(r)= r^k }, we have

    \displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} r^k \,dS\ge \int_{\partial \Omega^\#} r^k \,dS  \end{array}

    if {k\ge 1}. When {k=0}, this is just the classical isoperimetric inequality.

  2. On the hyperbolic space {\mathbb H^n}, the warping function is {s(r)=\sinh r}. Choosing {a(r)=b(r)=\sinh^k (r)}, we have

    \displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} \sinh^k r \,dS\ge \int_{\partial \Omega^\#} \sinh^k r \,dS  \end{array}

    if {k\ge 1}. Similarly we also have

    \displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega} \cosh r \,dS\ge \int_{\partial \Omega^\#} \cosh r \,dS\end{array}

    and

    \displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega} (\cosh r-1)^k \,dS\ge \int_{\partial \Omega^\#} (\cosh r-1)^k \,dS\end{array}

    if {k\ge 1}.

  3. On the open hemisphere {\mathbb S^{n}_+}, the warping function is {s(r)=\sin r}, {(0<r<\frac{\pi}{2})}. Choosing {a(r)=b(r)=\tan^k (r)}, we have

    \displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega} \tan^k r \,dS\ge \int_{\partial \Omega^\#} \tan^k r \,dS\end{array}

    if {k\ge 1} and {\Omega\subset \mathbb S^n_+}. Similarly we also have

    \displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega} (1-\cos r) \,dS\ge \int_{\partial \Omega^\#} (1-\cos r) \,dS.\end{array}

In all the above examples, we can convert the inequalities into a form which involves the volume of {\Omega}:

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega}a(r)dS\ge |N|a(R)A(R) \end{array}

where {R=V^{-1} (|\Omega|)}. For example, in {\mathbb R^n}, the inequality

\displaystyle \begin{array}{rl} \displaystyle \int_{\partial \Omega}r^k dS\ge \int_{\partial \Omega^\#}r^k dS\end{array}

is equivalent to

\displaystyle \begin{array}{rl} \displaystyle  \int_{\partial \Omega}r^k dS\ge n \beta_n ^{-\frac{k-1}{n}} \mathrm{Vol} (\Omega) ^{\frac{n-1+k}{n}} \ \ \ \ \ (11)\end{array}

for {k\ge 0}, where {\beta_n} is volume of the unit ball {B} in {\mathbb R^n}. For other spaces, the inequality is not as explicit because {V^{-1}} is not explicit except when {n=2}.

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 {H_0=1}.

From now on, our ambient space is the warped product manifold {M^{n} = [0,l)\times {N}^{n-1}} equipped with the metric {g=dr^2+s(r)^2 g_N}.

Before stating the main theorem, we give some definitions which are useful in studying the extrinsic geometry of hypersurfaces. On a hypersurface {\Sigma} in {M}, we define the normalized {k}-th mean curvature function

\displaystyle \begin{array}{rl} \displaystyle  \begin{array}{rcl}  H_k:=H_k(\Lambda)=\frac{1}{\binom{n-1}{k}}\sigma_k(\Lambda), \end{array} \end{array}

where {\Lambda=(\kappa_1,\cdots,\kappa_{n-1})} are the principal curvature functions on {\Sigma} and the homogenous polynomial {\sigma_k} of degree {k} is the {k}-th elementary symmetric function

\displaystyle \begin{array}{rl} \displaystyle  \sigma_k(\Lambda)=\sum_{i_1<\cdots<i_{k}}\kappa_{i_1}\cdots \kappa_{i_k}. \end{array}

We adopt the usual convention {\sigma_0=H_{0}=1}.

The {k}-th Newton transformation {T_k: T\Sigma \rightarrow T \Sigma} (cf. [R]) is useful in studying the extrinsic geometry of {\Sigma}, and is defined as follows. If we write

\displaystyle \begin{array}{rl} \displaystyle   T_k ( e_j ) =\sum_{i=1}^{n-1} ( T_k )_j^i e_{i},  \end{array}

then { (T_k) _j^i } are given by

\displaystyle \begin{array}{rl} \displaystyle  {(T_k)}_j^{\,i}= \frac 1 {k!} \sum_{\substack{1 \le i_1,\cdots, i_k \le n-1\\ 1\le j_1, \cdots, j_k \le n-1}} \delta^{i i_1 \ldots i_k}_{j j_1 \ldots j_k} B_{i_1}^{j_1}\cdots B_{i_k}^{j_k} \end{array}

where {B} is the second fundamental form of {\Sigma}. One also defines {T_0 = \mathrm{Id}}, the identity map.

We define the vector field {X = s(r) \, \frac{\partial}{\partial r}} and the potential function

\displaystyle \begin{array}{rl} \displaystyle   c(r) = s'(r),  \end{array}

which will be used to define the weighted volume {\mathrm{Vol}_c} ((9)). Note that {X} is a conformal Killing vector field: {\mathcal L_X g = 2 c g} [Br][Lemma 2.2].

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 {\eta} is a smooth function on a closed hypersurface {\Sigma} in {M}, then for {1\le k \le n-1}, we have

\displaystyle \begin{array}{rl} \displaystyle     \int_\Sigma \eta c H_{k-1}dS =& \displaystyle \int_\Sigma \eta H_{k} \langle X, \nu\rangle dS -\frac{1}{k{{n-1}\choose k}}\int_\Sigma \eta \left(\mathrm{div}_\Sigma T_{k-1}\right)(X^T)dS\\ & \displaystyle -\frac{1}{k{{n-1}\choose k}}\int_\Sigma \langle T_{k-1} (X^T), \nabla _\Sigma \eta\rangle dS,   \end{array}

where {\nu} is the unit normal vector, {X=s (r) \partial _r} and {X^T} is the tangential component of {X} onto {T\Sigma}.

Proof: For completeness we sketch the proof here. Let {m=n-1}. We compute

\displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \mathrm{div}_\Sigma \left(\eta T_k (X^T)\right)\\ =& \displaystyle \langle T_k(\nabla \eta), X^T\rangle+ \eta(\mathrm{div}_\Sigma\;T_k)(X^T)+ \frac{1}{2}\eta\langle T_k, \iota^* (\mathcal{L}_Xg )\rangle -\eta\langle T_k, B\rangle \langle X, \nu\rangle\\ =& \displaystyle \langle T_k(\nabla \eta), X^T\rangle+ \eta(\mathrm{div}_\Sigma\;T_k)(X^T)+ c \eta\langle T_k, \iota^*g \rangle -\eta\langle T_k, B\rangle \langle X, \nu\rangle\\ =& \displaystyle \langle T_k(\nabla \eta), X^T\rangle+ \eta(\mathrm{div}_\Sigma\;T_k)(X^T)+ (m-k){m\choose k}c \eta H_k -(k+1){m\choose {k+1}}H_{k+1}\eta \langle X, \nu\rangle \end{array}

where {\iota} is the inclusion of {\Sigma} in {M} and we used the fact that {\mathrm{tr}_\Sigma(T_k)= (m-k){m\choose k}H_k} and {\langle T_k, B\rangle =(k+1){m\choose {k+1}} H_{k+1}}. Applying the divergence theorem will then give the result. \Box

Lemma 15 Suppose {N} has constant curvature {K} and {\Sigma} is a star-shaped hypersurface with {H_p>0}. Assume that {s'(r)>0} for {r>0} and {s'(r)^2-s(r)s''(r)\le K}. Then

  1. For all {k \in \{1, \cdots, p-1\}}, we have {T_k>0} and {H_k>0}.
  2. For {k \in \{2, \cdots, p\}},

    \displaystyle \begin{array}{rl} \displaystyle  (\mathrm{div}_\Sigma \,T_{k-1}) (X^T)\ge 0, \ \ \ \ \ (12)\end{array}

    where {X^T} is the tangential component of {X} onto {T\Sigma}.

Proof:

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 {K>s'(r)^2-s(r)s''(r)} (Condition (H4) in [Br], [BE]) and {\langle X, \nu\rangle >0}, but since we only require non-strict inequality in (12), the conclusion still holds under our assumption.

A remark is that we need {N} to have constant curvature because conformal flatness of {g} is essential in the formula of {\left(\mathrm{div}_{\Sigma}T_{k-1}\right)(X^T)} on p. 393 in [BE].

\Box

Theorem 16 Suppose {\Omega} is a domain in {M} and its boundary is a smooth star-shaped hypersurface with {H_1 \ge 0}. Assume that {s'>0} for {r>0} and {0\le s'(r)^2-s(r)s''(r)}. Then for {l\ge 1},

\displaystyle \begin{array}{rl} \displaystyle   |N|^{-\frac{l-1}{n}} \left(n\int_{\Omega}c(r) dv\right)^{\frac{n+l-1}{n}} \le \int_{\partial \Omega} H_1 s(r)^{l+1} c(r)^{-1} \,dS. \end{array}

The equality holds if and only if {\partial \Omega} is a slice.

Proof: We will prove the following chain of inequalities:

\displaystyle \begin{array}{rl} \displaystyle    |N|^{-\frac{l-1}{n}} \left(n\int_{\Omega}c(r) dv\right)^{\frac{n+l-1}{n}} \le \int_{\partial \Omega} s(r)^{l} dS \le \int_{\partial \Omega} H_1 s(r)^{l+1} {c(r)}^{-1} \,dS.   \end{array}

Define {a(r)=s(r)^l }. As {A(r)=s(r)^{n-1}}, {\psi(r)=s(r)^{n-1+l} } and {\widetilde v(r)=\int_{0}^{r}s(t)^{n-1} c (t) dt=\frac{1}{n} s(r)^n}, we have

\displaystyle \begin{array}{rl} \displaystyle   \psi\circ \widetilde v^{-1} (u) =& \displaystyle  \left( n u\right)^{\frac{n-1+l}{n}} \end{array}

which is clearly convex as {l\ge 1}. So by Theorem 12, we have

\displaystyle \begin{array}{rl} \displaystyle  \int_{\partial \Omega}s(r)^l dS\ge \int_{\partial \widetilde \Omega^\#}s(r) ^l dS =|N|^{-\frac{l-1}{n}} \left(n\int_{\Omega}c(r) dv\right)^{\frac{n+l-1}{n}}. \ \ \ \ \ (13)\end{array}

We now simply denote {s(r)} by {s} and {c(r)} by {c}. Applying the weighted Hsiung-Minkowski formula (Proposition 14), we have

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} s^l dS =\int_{\partial \Omega} c\frac{s^l}{c} dS =& \displaystyle \int_{\partial \Omega} \frac{s^l}{c} H_1 \langle X, \nu \rangle-\frac{1}{n-1} \int_{\partial \Omega} \left\langle \nabla \left(\frac{s^l}{c}\right), s \nabla r\right\rangle dS\\ =& \displaystyle \int_{\partial \Omega} \frac{s^l}{c} H_1 \langle X, \nu \rangle-\frac{1}{n-1} \int_{\partial \Omega} \left\langle \frac{s^{l-1} }{c^2} (lc^2-ss'')\nabla r, s \nabla r\right\rangle dS\\ \le& \displaystyle \int_{\partial \Omega} \frac{s^l}{c} H_1 \langle X, \nu \rangle dS\\ \le& \displaystyle \int_{\partial \Omega} \frac{s^{l+1} }{c} H_1 dS. \end{array}

By Theorem 12, the equality holds if and only if {\partial \Omega} is a slice. \Box

To relate the weighted volume to the integral of the higher order mean curvatures, we need stronger assumptions.

Theorem 17 Suppose {N} has constant curvature {K} and {\Sigma} is a closed star-shaped hypersurface which is the boundary of a domain {\Omega}. Assume that {s'>0} for {r>0}, {0\le s'(r)^2-s(r)s''(r)\le K} and {H_k>0} on {\Sigma}. Then for {l\ge 1},

\displaystyle \begin{array}{rl} \displaystyle   |N|^{-\frac{l-1}{n}} \left(n\int_{\Omega}c(r) \, dv\right)^{\frac{n+l-1}{n}} \le \int_{\partial \Omega} H_k s(r)^{l+k} c(r)^{-k} \,dS. \end{array}

The equality holds if and only if {\Sigma} is a slice.

Proof: We actually prove the following stronger statement:

\displaystyle \begin{array}{rl} \displaystyle   |N|^{-\frac{l-1}{n}} \left(n\int_{\Omega}c(r) dv\right)^{\frac{n+l-1}{n}} \le \int_{\partial \Omega} s(r)^{l} dS \le& \displaystyle  \int_{\partial \Omega} H_1 s(r)^{l+1} c(r)^{-1} dS\\ \le& \displaystyle  \cdots\\ \le& \displaystyle  \int_{\partial \Omega} H_k s(r)^{l+k} c(r)^{-k} dS.  \ \ \ \ \ (14)\end{array}

The first two inequalities have already been proved in Theorem 16.

We prove the remaining inequalities by induction. Again denote {s(r)} by {s} and {c(r)} by {c}. By the weighted Hsiung-Minkowski formula (Proposition 14) and Lemma 15, for {1\le j\le k-1}, we have

\displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \int_{\partial \Omega} \frac{s^{l+j}}{c ^j }H_j dS\\ =& \displaystyle \int_{\partial \Omega} \frac{s^{l+j} }{c^{j+1} } H_{j+1} \langle X, \nu \rangle dS -\frac{1}{j{{n-1}\choose j}} \int_{\partial \Omega} \left\langle \frac{s^{l+j-1} }{c ^{j+2} } ((l+j)c^2 -(j+1)ss'' )T_j(\nabla r), s \nabla r\right\rangle dS\\ \le& \displaystyle \int_{\partial \Omega} \frac{s^{l+j} }{c^{j+1} } H_{j+1} \langle X, \nu \rangle dS\\ \le& \displaystyle \int_{\partial \Omega} \frac{s^{l+j+1} }{c^{j+1} } H_{j+1}dS. \end{array}

By Theorem 16, the equality holds if and only if {\partial \Omega} is a slice. \Box

We note that the conditions in Theorem 16 and Theorem 17 are satisfied in the following space forms:

  1. The Euclidean space {\mathbb R^n} with metric {dr^2+r^2 \,g_{\mathbb S^{n-1}}}.
  2. The hyperbolic space {\mathbb H^n} with metric {dr^2+\sinh^2 r \, g_{\mathbb S^{n-1}}}.
  3. The open hemisphere {\mathbb S^n_+} with metric {dr^2+\sin^2 r \, g_{\mathbb S^{n-1}}}.

In the following corollaries, we denote the point {\{r=0\}} by {0} and the volume of the unit ball in {\mathbb R^n} by {\beta_n}. We obtain the following corollaries.

Corollary 18 Let {\Sigma} be a closed embedded hypersurface in {\mathbb R^{m+1}} which is star-shaped with respect to {0} and {\Omega} is the region enclosed by it. Assume that {H_k>0} on {\Sigma}. Then for any integer {l\ge 0},

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n^{-\frac{l-1}{n}}\mathrm{Vol} (\Omega)^{\frac{n-1+l}{n}} \le\int_{\Sigma} H_kr^{l+k}dS. \end{array}

Proof: The case where {l\ge 1} follows directly from Theorem 17. If {k=l=0}, this is the ordinary isoperimetric inequality. So (13) is still true when {l=0}, and we can perform induction starting from this case to show the assertion when {l= 0}. \Box

Remark 2 If {l=1}, the inequality becomes

\displaystyle \begin{array}{rl} \displaystyle   n\mathrm{Vol} (\Omega) \le \int _{\partial \Omega} H_kr^{k+1}dS. \end{array}

In particular, if {k=1}, it is easily seen from the proof that the assumption can be weakened to {H_1\ge 0} because {T_0=\mathrm{Id}} is always positive. Corollary 18 extends (11) in Section 4 and also generalizes [KM] Theorem 2. See also [KM2][Theorem 2].

Corollary 19 Let {\Sigma} be a closed embedded hypersurface in {\mathbb H^{m+1}} which is star-shaped with respect to {0} and {\Omega} is the region enclosed by it. Assume that {H_k>0} on {\Sigma}. Then for any integer {l\ge 1},

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n^{-\frac{l-1}{n}} \left(\int_{\Omega}\cosh r \,dv\right)^{\frac{n+l-1}{n}} \le \int_{\partial \Omega} H_k \sinh^l r\tanh ^k r \,dS. \end{array}

Corollary 20 Let {\Sigma} be a closed embedded hypersurface in {\mathbb S_+^{m+1}} which is star-shaped with respect to {0} and {\Omega} is the region enclosed by it. Assume that {H_k>0} on {\Sigma}. Then for any integer {l\ge 1},

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n^{-\frac{l-1}{n}} \left(\int_{\Omega}\cos r \, dv\right)^{\frac{n+l-1}{n}} \le \int_{\partial \Omega} H_k \sin^l r\tan ^k r \,dS. \end{array}

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

In this section, we apply our isoperimetric results to obtain some sharp eigenvalue estimates, a Pólya-Szegö inequality, a Faber-Krahn inequality and a Cheeger-type eigenvalue theorem.

First, we give an upper bound for the first eigenvalue of a differential operator related to the Newton’s tensor. We define {\lambda_1(T_k)} to be the first eigenvalue of the symmetric second order differential operator {-\mathrm{div} (T_k\circ\nabla )} on {\Sigma}. The equality holds if and only if {\Sigma} is immersed as a geodesic sphere. Note that {\lambda_1(T_0)} 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 {k=0}.

Theorem 21 Let {\Sigma} be a closed embedded hypersurface in {\mathbb R^{m+1}} enclosing a region {\Omega}. Then

\displaystyle \begin{array}{rl} \displaystyle   \lambda_1(T_k)\le \frac{(m-k){m\choose k}{\beta_n}^{\frac{1}{n}}}{n \mathrm{Vol} (\Omega)^{\frac{n+1}{n}}}\int_{\partial \Omega} H_k dS \end{array}

where {n=m+1} and {\beta_n} is the volume of the unit ball in {\mathbb R^n}. The equality holds if and only if {\Sigma} is a hypersphere. (Note that {\lambda_1(T_0)} is just the first Laplacian eigenvalue.)

Proof: By a suitable translation, we can assume that {\int_{\partial \Omega}x_i\,dS=0} for {i=1, \cdots, n}.

By Theorem 3, we have

\displaystyle \begin{array}{rl} \displaystyle  \int_{\partial \Omega} r^2 dS \ge \int_{\partial \Omega^\#}r^2 dS = n \beta_n^{-\frac{1}{n}} \mathrm{Vol} (\Omega)^{\frac{n+1}{n}}. \ \ \ \ \ (15)\end{array}

By the variational characterization of {\lambda_1(T_k)} and the fact that {\textrm{tr}_\Sigma(T_k)=(m-k)H_k}, we have

\displaystyle \begin{array}{rl} \displaystyle    \lambda_1(T_k) \int_{\partial \Omega} r^2 =& \displaystyle \lambda_1(T_k)\sum_{i=1}^n \int_{\partial \Omega} {x_i} ^2 \le \int_{\partial \Omega} \sum_{i=1}^n \langle T_k (\nabla x_i), \nabla x_i\rangle dS\\ =& \displaystyle \int_{\partial \Omega}\sum_{j,l=1}^m \left(\sum_{i=1}^n(\nabla _{e_j}x_i)(\nabla _{e_l}x_i) \right)(T_k)_j^l dS\\ =& \displaystyle \int_{\partial \Omega} \mathrm{tr}_\Sigma(T_k) dS\\ =& \displaystyle \int_{\partial \Omega} (m-k){m\choose k}H_k dS.   \end{array}

Therefore combining the two inequalities we have

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n^{-\frac{1}{n}} \mathrm{Vol} (\Omega)^{\frac{n+1}{n}} \lambda_1(T_k) \le (m-k){m\choose k}\int_\Sigma H_k dS. \end{array}

If the equality holds, then by Theorem 3, {\partial \Omega} is a hypersphere. \Box

Corollary 22 Let {\Sigma} be a closed embedded hypersurface in {\mathbb R^{m+1}} enclosing a region {\Omega}. Then the first Laplacian eigenvalue {\lambda_1(\Sigma)} on {\Sigma} satisfies

\displaystyle \begin{array}{rl} \displaystyle   \lambda_1(\Sigma)\le \frac{m\beta_n^{\frac{1}{n}} \mathrm{Area} ( \Sigma)}{n\mathrm{Vol} (\Omega)^{\frac{n+1}{n}}}. \end{array}

The equality is attained if and only if {\Omega} is a ball.

To state our next result, we need to define the Steklov eigenvalues, as follows. Let {(\Omega,g)} be a compact Riemannian manifold with smooth boundary {\partial \Omega=\Sigma}. The first nonzero Steklov eigenvalue is defined as the smallest {p\ne0} of the following Steklov problem ([St])

\displaystyle \begin{array}{rl} \displaystyle  \begin{cases} \Delta f =0\quad & \displaystyle \textrm{on }\Omega\\ \frac{\partial f}{\partial \nu}=p f \quad & \displaystyle \textrm{on }\partial \Omega \end{cases} \ \ \ \ \ (16)\end{array}

where {\nu} is the unit outward normal of {\partial \Omega}. Physically, this describes the stationary heat distribution in a body {\Omega} whose flux through {\partial \Omega} is proportional to the temperature on {\partial \Omega}. It is known that the Steklov boundary problem (16) has a discrete spectrum

\displaystyle \begin{array}{rl} \displaystyle 0=p_0< p_1\le p_2\le\cdots \rightarrow \infty.\end{array}

Moreover, {p_1} has the following variational characterization (e.g. [KS] [Equation 2.3])

\displaystyle \begin{array}{rl} \displaystyle  p_1(\Omega)=\min \left\{\frac{\int_\Omega|\nabla f|^2 dv}{\int_{\partial \Omega}f^2 dS}\left|\, \int_{\partial \Omega}f dS=0\right\}.\right. \ \ \ \ \ (17)\end{array}

We will now prove an upper bound of {p_1(\Omega)} with the techniques similar to that in Theorem 21.

Theorem 23 Suppose {\Omega} is a domain in the hyperbolic space {\mathbb H^n} with smooth boundary, then the first Steklov eigenvalue {p_1} satisfies

\displaystyle \begin{array}{rl} \displaystyle   p_1(\Omega)\le \frac{n\mathrm{Vol} (\Omega)+\int_\Omega \sinh^2 r\,dv}{n \beta_n^{-\frac{1}{n}} \left(\int_\Omega \cosh r\,dv\right)^{\frac{n+1}{n}}}.  \end{array}

after possibly a translation of the origin.

Proof: We use the hyperboloid model for {(\mathbb H^n, g)}:

\displaystyle \begin{array}{rl} \displaystyle \mathbb H^n=\lbrace x\in \mathbb R^{n,1}:(x_0, x_1,\cdots, x_n)= (\cosh r , (\sinh r )\theta ),\; r\geq 0, \;\theta \in \mathbb S^{n-1}\rbrace\end{array}

with metric induced from the Minkowski metric {\sum_{i=1}^n (dx_i)^2-(dx_0)^2}. Intrisically, {g=dr^2+\sinh^2 r g_{\mathbb S^{n-1}}}. By applying a rigid motion of {\mathbb H^n}, we can assume that for {i=1, \cdots, n},

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega} x_i dS=0.  \end{array}

So by (15) and (17),

\displaystyle \begin{array}{rl} \displaystyle   n \beta_n^{-\frac{1}{n}} \left(\int_\Omega \cosh rdv\right)^{\frac{n+1}{n}} =\int_{\partial \Omega^\# } \sinh^2 r dS \le& \displaystyle  \int_{\partial \Omega} \sinh ^2 r dS\\ =& \displaystyle  \sum_{i=1}^{n}\int_{\partial \Omega} {x_i}^2 dS\\ \le& \displaystyle  \sum_{i=1}^{n}\frac{1}{p_1} \int_{\Omega}|\nabla x_i|^2 dv\\ =& \displaystyle \frac{1}{p_1} \int_{\Omega}\left(n+|\nabla x_0|^2\right) dv\\ =& \displaystyle \frac{1}{p_1} \int_{\Omega}\left(n+\sinh ^2 r \right) dv\\ =& \displaystyle \frac{1}{p_1} \left(n \mathrm{Vol} (\Omega)+\int_{\Omega}\sinh ^2 r dv\right). \end{array}

From this we obtain

\displaystyle \begin{array}{rl} \displaystyle   p_1(\Omega)\le \frac{n\mathrm{Vol} (\Omega)+\int_\Omega \sinh^2 rdv}{n \beta_n^{-\frac{1}{n}} \left(\int_\Omega \cosh rdv\right)^{\frac{n+1}{n}}}.  \end{array}

\Box

Remark 3 Theorem 23 is the hyperbolic analogue of the corresponding result for {\Omega\subset \mathbb R^n}: {p_1(\Omega)\le \left(\frac{\beta_n}{\mathrm{Vol} (\Omega)}\right)^{\frac{1}{n}}}, which is implied by [Brock] [Theorem 3].

We now prove a Pólya-Szegö inequality. For a function {f} on {\mathbb R^n}, the Schwarz symmetrization ([S]) of {f} is the non-increasing radial function {f^\#} such that for all {t},

\displaystyle \begin{array}{rl} \displaystyle   \mathrm{Vol} \{|f|> t\} = \mathrm{Vol} \{f^\# >t\}.  \end{array}

The classical Pólya-Szegö inequality then states that

\displaystyle \begin{array}{rl} \displaystyle   \int_{\mathbb R^n}|\nabla f|^p dv\ge \int_{\mathbb R^n}|\nabla f^\#|^p dv \end{array}

if {f\in W^{1,p} (\mathbb R^n)}, where {p\ge 1}.

Theorem 24 With the same assumptions as Theorem 3, and assume {\Phi} is a convex non-decreasing function on {[0, \infty)}. Then for any Lipschitz function {f} on {M}, we have

\displaystyle \begin{array}{rl} \displaystyle   \int_M \Phi\left(a(r)|\nabla f|\right)dv_g \ge \int_M \Phi\left(a(r)|\nabla f^\#|\right)dv_g \end{array}

whenever the integral on the left is defined.

Proof: For simplicity, let us assume {f\in C^1_c(M, [0, \infty))}. By abuse of notation, we sometimes regard {f^\#=f^\#(r)}. Define {f^*} by {f^*(V(r))=f^\#(r)}. We claim that if {0<u<|\mathrm{supp}(f)|},

\displaystyle \begin{array}{rl} \displaystyle   \frac{d}{du} \left(\int_{\{x:f(x)>f^*(u)\}} \Phi( a(r) |\nabla f|)dv_g\right) \ge \Phi\left(\frac{d}{du}\left( \int_{\{x:f(x)>f^*(u)\}} a(r) |\nabla f| dv_g\right)\right).  \ \ \ \ \ (18)\end{array}

Note that if {|\mathrm{supp}(f)|-u>h>0}, {\mathrm{Vol} (\{x: f^*(u)\ge f(x)>f^*(u+h)\})=h} and so by Jensen’s inequality,

\displaystyle \begin{array}{rl} \displaystyle   & \displaystyle \frac{1}{h} \left[ \int_{\{x: f(x)>f^*(u+h)\}} \Phi( a(r) |\nabla f|)dv_g - \int_{\{x: f(x)>f^*(u)\}} \Phi( a(r) |\nabla f|)dv_g \right] \\ = & \displaystyle \frac{1}{h} \int_{\{x: f^*(u)\ge f(x)>f^*(u+h)\}} \Phi( a(r) |\nabla f|)dv_g \\ \ge& \displaystyle \Phi\left(\frac{1}{h} \int_{\{x: f^*(u)\ge f(x)>f^*(u+h)\}} a(r) |\nabla f| dv_g \right). \end{array}

The case where {h<0} is similar. Letting {h\rightarrow 0}, we get (18). On the other hand, by the coarea formula,

\displaystyle \begin{array}{rl} \displaystyle   \frac{d}{du}\left( \int_{\{x: f(x)>f^*(u)\}} a(r) |\nabla f| dv_g\right) =& \displaystyle -\left(\int _{\{x: f(x)=f^*(u)\}}a(r) \,dS\right) (f^*)'(u).  \ \ \ \ \ (19)\end{array}

By definition, {\mathrm{Vol} \{x: f(x)> f^*(u)\} =\mathrm{Vol} \{x: f^\#(x)> f^*(u)\} } and so by the weighted isoperimetric inequality (Theorem 3),

\displaystyle \begin{array}{rl} \displaystyle   \int _{\{x: f(x)=f^*(u)\}}a(r) \,dS \ge \int _{\{x: f^\#(x)=f^*(u)\}}a(r) \,dS \end{array}

So in view of (19), we have

\displaystyle \begin{array}{rl} \displaystyle   \frac{d}{du}\left( \int_{\{x: f(x)>f^*(u)\}} a(r) |\nabla f| dv_g\right) =& \displaystyle -\left(\int _{\{x: f(x)=f^*(u)\}}a(r) \,dS\right) (f^*)'(u)\\ \ge& \displaystyle -\left(\int _{\{x: f^\#(x)=f^*(u)\}}a(r) \,dS\right) (f^*)'(u)\\ =& \displaystyle -a(r)|N|A(r)\frac{(f^\#)'(r)}{|N|A(r)}\\ =& \displaystyle  -a(r)(f^\#)'(r)\\ =& \displaystyle  a(r) \left|\nabla f^\#\left(V^{-1} (u)\right)\right|. \end{array}

where {V(r)=u}, noting that {(f^*)'\le 0}. Combining this with (18), we have

\displaystyle \begin{array}{rl} \displaystyle   \frac{d}{du} \left(\int_{\{f>f^*(u)\}} \Phi( a(r) |\nabla f|)dv_g\right) \ge& \displaystyle  \Phi\left(\frac{d}{du}\left( \int_{\{f>f^*(u)\}} a(r) |\nabla f| dv_g\right)\right)\\ \ge& \displaystyle  \Phi\left(a(r) \left |\nabla f^\#\left(V^{-1} (u)\right)\right| \right). \end{array}

Integrating the above on {[0, \mathrm{Vol} (M))} and using a change of variable {u=V(r)}, we obtain

\displaystyle \begin{array}{rl} \displaystyle   \int_M \Phi\left( a(r) |\nabla f|\right)dv_g \ge& \displaystyle \int_0^{\mathrm{Vol} (M)} \Phi\left(a(r) \left|\nabla f^{\#}\left(V^{-1} (u)\right) \right| \right)du\\ =& \displaystyle \int_M \Phi\left(a(r) |\nabla f^{\#}|\right)dv_g. \end{array}

\Box

As an application of Theorem 24, we prove a Faber-Krahn inequality for the {p}-Laplacian. Let {\Omega} be a bounded region with smooth boundary sitting inside the warped product manifold {([0, l)\times \mathbb S^m, dr^2+s(r)^2g_{\mathbb S^m})}, where {g_{\mathbb S^m}} is the standard round metric. We assume {s(0)=0} and {s'(0)=1} and regard {\{r=0\}} as a point.

We consider the {p}-Laplacian eigenvalue problem on {\Omega} under the Dirichlet boundary condition. Namely, for {p>1}, we consider the following equation:

\displaystyle \begin{array}{rl} \displaystyle   \Delta _p f:=\mathrm{div}\left(|\nabla f|^{p-2}\nabla f\right)=-\lambda |f|^{p-2}f\;\textrm{on }\Omega, \quad f=0 \;\textrm{on }\partial \Omega,  \ \ \ \ \ (20)\end{array}

which is understood to hold in the weak sense. This is the Euler-Lagrange equation of the {p}-Dirichlet integral

\displaystyle \begin{array}{rl} \displaystyle \frac{1}{p}\int_\Omega |\nabla f|^p dv_g.\end{array}

The study of the {p}-Laplacian is of interest in the theory of non-Newtonian fluids (cf. [L]). The smallest eigenvalue {\lambda_{1,p} (\Omega)} is the smallest positive {\lambda} such that (20) holds for some non-trivial {f}. It is known that

\displaystyle \begin{array}{rl} \displaystyle   \lambda_{1,p} (\Omega) =\inf\left\{ \frac{\int_\Omega |\nabla f|^p dv_g}{\int_\Omega |f| ^p dv_g}: 0\ne f\in W_ 0 ^{1,p} (\Omega)\right\}.  \ \ \ \ \ (21)\end{array}

Theorem 25 Let { M=[0, l)\times \mathbb S^{m}}, equipped with the metric {g=dr^2+s(r)^2g_{\mathbb S^m}}. Suppose the classical inequality holds on {(M,g)} and that {s(0)=0}, {s'(0)=1}. Then for a domain {\Omega} in {M} with smooth boundary, we have

\displaystyle \begin{array}{rl} \displaystyle   \lambda_{1,p} (\Omega)\ge \lambda_{1,p} (\Omega^\#)  \end{array}

for {p> 1.}

Proof: Our assumptions ensure that {g} is sufficiently smooth at {0}. Suppose {f} is the first eigenfunction on {\Omega}, which we can assume to be non-negative, and let {f^{\#}} be its Schwarz symmetrization. Then by Fubini’s theorem,

\displaystyle \begin{array}{rl} \displaystyle   \int_\Omega f ^p dv_g =\int_0^\infty \mathrm{Vol}\{ f ^p >t \}dt =\int_0^\infty \mathrm{Vol}\{ f >t^{\frac{1}{p}} \}dt =& \displaystyle \int_0^\infty \mathrm{Vol}\{ f^\# >t^{\frac{1}{p}} \}dt\\ =& \displaystyle \int_0^\infty \mathrm{Vol}\{ ({f^\#})^p > t \}dt\\ =& \displaystyle \int_{\Omega^\#} ( f^\# )^p dv_g.  \ \ \ \ \ (22)\end{array}

By Theorem 24, we have

\displaystyle \begin{array}{rl} \displaystyle   \int_\Omega|\nabla f|^p dv_g\ge \int_{\Omega^\#}|\nabla f^\#|^p dv_g.  \ \ \ \ \ (23)\end{array}

By (22), (23) and the minimax principle (21) ,

\displaystyle \begin{array}{rl} \displaystyle   \lambda_{1,p} (\Omega) =\frac{\int_\Omega|\nabla f|^p dv_g}{\int_\Omega f ^p dv_g} \ge\frac{\int_\Omega|\nabla f^\#|^p dv_g}{\int_{\Omega^\#} (f^\#) ^p dv_g} \ge\lambda_{1,p} (\Omega^\#). \end{array}

\Box

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 {\Omega} is a Riemannian manifold with smooth non-empty boundary. Consider the weighted Laplacian eigenvalue problem in divergence form:

\displaystyle \begin{array}{rl} \displaystyle   L(f):= \mathrm{div}\left(a(x)^2\,\nabla f\right) =-\lambda f \;\textrm{on }\Omega, \quad f=0 \;\textrm{on }\partial \Omega \end{array}

where {a(x)} is a positive weighted function on {\Omega}. Then

\displaystyle \begin{array}{rl} \displaystyle   \lambda_1(L)=\inf\left\{\frac{\int_\Omega a(x)^2|\nabla f|^2dv}{\int_\Omega f ^2dv}: 0\ne f\in W^{1,2}_0(\Omega)\right\}. \ \ \ \ \ (24)\end{array}

We define the weighted Cheeger’s constant to be

\displaystyle \begin{array}{rl} \displaystyle   h_a(\Omega):= \inf \left\{ \frac{\int_{\partial \Omega_0}a(x)dS }{\mathrm{Vol}(\Omega_0)}: \Omega_0\Subset \Omega \right\}. \end{array}

Theorem 26 For a compact domain {\Omega} with boundary,

\displaystyle \begin{array}{rl} \displaystyle   \lambda_1(L)\ge \frac{1}{4}h_a(\Omega)^2. \end{array}

Proof: We suspect this result is known to experts. We include the proof here since we cannot find an explicit reference. First, suppose {\phi} is a non-negative function on {\Omega} with {\phi|_{\partial \Omega}=0}, then by the coarea formula and Fubini’s theorem,

\displaystyle \begin{array}{rl} \displaystyle    \int_\Omega a(x)|\nabla \phi|dv =& \displaystyle \int_{0}^\infty \int_{\{\phi=t\}} a(x)dS \,dt\\ =& \displaystyle  \int_{0}^\infty \frac{\int_{\{\phi=t\}} a(x)dS }{\mathrm{Vol}\{\phi>t\}} \cdot\mathrm{Vol}\{\phi>t\}dt\\ \ge& \displaystyle  \inf _{s}\left(\frac{\int_{\{\phi=s\}} a(x)dS }{\mathrm{Vol}\{\phi>s\}}\right) \int_{0}^\infty \mathrm{Vol}\{\phi>t\}dt\\ =& \displaystyle  h_a(\Omega)\int_\Omega \phi \,dv.  \ \ \ \ \ (25)\end{array}

Let {f} be the first eigenfunction, which we can assume to be non-negative. Take {\phi=f^2} in (25), then

\displaystyle \begin{array}{rl} \displaystyle   h_a(\Omega)\int_\Omega f^2dv \le& \displaystyle  \int_\Omega a(x)|\nabla (f^2)|dv\\ =& \displaystyle 2\int_\Omega a(x)|f||\nabla f |dv\\ \le& \displaystyle 2 \left(\int_\Omega f^2 dv\right)^{\frac{1}{2}} \left(\int_\Omega a(x)^2|\nabla f |^2dv\right)^{\frac{1}{2}}. \end{array}

From this and (24), we get {\lambda_1(L)\ge \frac{1}{4}h_a(\Omega)^2}. \Box

It is not always easy to compute the (weighted) Cheeger’s constant. However, in the warped product setting and when {a=a(r)} is a radial function (but without assuming {\Omega} is symmetric), it can be readily estimated by the following result.

Theorem 27 Let { M=[0, l)\times \mathbb S^{m}}, equipped with the metric {g=dr^2+s(r)^2g_{\mathbb S^m}} with {s(0)=0} and {s'(0)=1}. Suppose {(M,g)} and {a(r)} satisfies the assumptions of Theorem 8, and assume {\Omega} is a bounded domain in {M} with smooth boundary. Then

\displaystyle \begin{array}{rl} \displaystyle   h_a(\Omega)\ge \inf_{0<u<\mathrm{Vol}(\Omega)} \frac{\Phi(u)}{u}. \end{array}

where {\Phi(u)=a(V^{-1}(u))A(V^{-1}(u))}. In particular,

\displaystyle \begin{array}{rl} \displaystyle   \lambda_1(L)\ge \frac{1}{4}\left(\inf_{0<u<\mathrm{Vol}(\Omega)} \frac{\Phi(u)}{u}\right)^2.  \end{array}

Proof: By Theorem 8, we have

\displaystyle \begin{array}{rl} \displaystyle   \int_{\partial \Omega_0} a(r)dS \ge& \displaystyle \int_{\partial \Omega_0^\#} a(r)dS =\Phi(\mathrm{Vol}(\Omega_0)). \end{array}

We conclude that

\displaystyle \begin{array}{rl} \displaystyle   h_a(\Omega)\ge \inf_{0<u<\mathrm{Vol}(\Omega)} \frac{\Phi(u)}{u}. \end{array}

\Box

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 {A\circ V^{-1}} being a convex function (Assumption 1), or equivalently that {ss''-s'^2\ge 0} if {s} is twice differentiable. We will show that this condition is necessary in the following sense:

Proposition 28 Suppose {s(r_0)s''(r_0)-s'(r_0)^2<0}, then there exists a compact Riemannian manifold {(N, g_N)} such that for the warped product manifold {([0, l)\times N, dr^2+s(r)^2g_N)}, the coordinate slice {\Sigma=\{r=r_0\}} 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 {\Sigma}, 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)])

\displaystyle \begin{array}{rl} \displaystyle   \left.\frac{d^2}{dt^2}\right|_{t=0}\mathrm{Area} (\Sigma_t) = \int_{\Sigma} \left(|\nabla_\Sigma u|^2-\left(|B|^2+\mathrm{Ric}_g (\nu, \nu)\right)u^2\right)dS \end{array}

where the 1-parameter family of deformations of {\Sigma} is given by {\Sigma_t=\phi(\Sigma, t)} with {\Sigma_0=\Sigma}, {|B|^2} is the square norm of the second fundamental form, {u} is a function on {\Sigma} and the normal variation is {u\nu_{\Sigma_t}}. In order to preserve the volume, we require {\int_\Sigma u dS=0}.

Suppose {s(r_0)s''(r_0)-s'(r_0)^2=-k<0}. Choose a compact Riemannian manifold {(N, g_N)} such that its first Laplacian eigenvalue {\lambda_1(g_N)<m k} and let {u} be the corresponding eigenfunction. Such {N} can, for example, be a sphere {\mathbb S^m(R)} with sufficiently large radius {R} such that {\frac{1}{R^2}<k}.

Let {M=[0, l)\times N} equipped with the metric {g=dr^2+s(r)^2 g_N}, then its straightforward to compute that

\displaystyle \begin{array}{rl} \displaystyle   |B|^2+\mathrm{Ric}_g(\nu, \nu)=m\left(\frac{s'(r_0)^2-s(r_0)s''(r_0)}{s(r_0)^2}\right)  \end{array}

on the hypersurface {\Sigma=\{r=r_0\}}. It is also easy to see that {u} is an eigenfunction on {\Sigma} with eigenvalue {\frac{\lambda_1(g_N)}{s(r_0)^2}}, so by the variational characterization of {\lambda_1}, {u} satisfies {\int_\Sigma u dS=0} and

\displaystyle \begin{array}{rl} \displaystyle   \int_\Sigma |\nabla_\Sigma u|^2 dS=\frac{\lambda_1(g_N)}{s(r_0)^2}\int_\Sigma u^2 dS. \end{array}

Therefore the second variation formula becomes

\displaystyle \begin{array}{rl} \displaystyle   \left.\frac{d^2}{dt^2}\right|_{t=0}\mathrm{Area} (\Sigma_t) =& \displaystyle  \int_{\Sigma} \left(|\nabla_\Sigma u|^2-\left(|B|^2+\mathrm{Ric}_g (\nu, \nu)\right)u^2\right)dS\\ =& \displaystyle  \left(\lambda_1(g_N)-m\left(s'(r_0)^2-s(r_0)s''(r_0)\right)\right)\int_{\Sigma} \frac{u^2}{s(r_0)^2} dS\\ <& \displaystyle 0. \end{array}

It follows that {\Sigma} fails to be area-minimizing among nearby hypersurfaces enclosing the same volume. \Box

Remark 4 The necessity of the condition {ss''-s'^2\ge -1} when {N} 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 {-k\le ss''-s'^2\le 0}. Proposition 28 does not contradict the result in [GLW] because it is assumed that {\mathrm{Ric}_N \ge (m-1)k g_N} in [GLW] while in our example, {\mathrm{Ric}_N\ge (m-1)k g_N} cannot hold. Indeed, as already noted in [GLW], the condition that {\mathrm{Ric}_N \ge (m-1)k g_N} guarantees that {\lambda_1(g_N)\ge m k} by Lichnerowicz’s theorem [Li].

It is also easy to see that the condition that the surjectivity (Assumption 1) of the projection map {\pi: \partial \Omega\rightarrow N} is necessary if {s(0)>0}, or equivalently, {A(0)>0}. Indeed, if {A(0)>0}, {|B_r|\rightarrow 0} but {\mathrm{Area} (\{r=r_0\})\rightarrow A(0)} as {r\rightarrow 0^+}. If we take {\Omega} to be a small enough geodesic ball around a point which is far from {r=0}, then the isoperimetric inequality clearly fails as {\partial \Omega} has smaller area than {\partial B_r}, where {B_r=\Omega^\#}.

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 {[1, \infty)} and {N} to be any compact {m}-dimensional manifold. Let {s(r)=r^{-\frac{1}{m}}} which is a decreasing function. As {\log s(r)=-\frac{1}{m}\log r} is convex, (1) is satisfied. For the region {\Omega=\{R_1<r<R_2\}}, {|\Omega| =|N|(\log R_2 -\log R_1)} and { |\partial \Omega| = |N|({R_1}^{-1}+{R_2}^{-1})}. If we take {R_2=e R_1} and let {R_1\rightarrow \infty}, then we have {|\Omega|\equiv |N|} but {|\partial \Omega|\rightarrow 0}. Therefore the isoperimetric inequality fails.

Advertisements
This entry was posted in Analysis, Calculus, Differential equations, Functional analysis, Geometry, Inequalities. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s