Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces

Let $\Omega$ be a bounded domain with smooth boundary, and we consider the following Neumann eigenvalue problem

where $\nu$ is the outward unit normal to $\partial\Omega$. It is well known that the spectrum of problem (1.1) is discrete and denoted by

with its multiplicity. For a bounded domain in $\mathbb{R}^{n}$, the classic Szegö-Weinberger inequality asserts that the ball is the unique maximizer of the first nonzero Neumann eigenvalue among domains having the same volume in $\mathbb{R}^{n}$, namely,

where $B$ is a round ball in $\mathbb{R}^{n}$ with volume equal to $|\Omega|$ (the volume of $\Omega$). This result was proved by Szegö [10] in planer domains, and by Weinberger [13] for all dimensions. Moreover inequality (1.2) also holds for bounded domains in hyperbolic space, a hemisphere [4, 15] and $94\%$ of the $2$-sphere [8]. Recently, the second named author [11] proved a Szegö-Weinberger type inequality for bounded domains in curved spaces.

For bounded simply connected planer domains, applying Szegö’s conformal map technique one can prove

where $B$ is a round disk with $|B|=|\Omega|$. Moreover estimate (1.3) is sharp and the equality holds on round balls. In 1993, Ashbaugh and Benguria [3] improved (1.3) by removing the assumption of simply connectedness, and conjectured that

We refer to [2, 7] and references therein for some progress towards this conjecture. Recently, Wang and Xia [14] made an important progress, which supports Conjecture 1.1 well. Precisely, they proved

Wang and Xia [14] also proved that inequality (1.4) holds true for bounded domains in hyperbolic space; Benguria, Brandolini and Chiacchio proved (1.4) for bounded domains in a hemisphere [5] later. Besides, Wang and Xia [12] proved a similar result as (1.4) for the ratio of Dirichlet eigenvalues.

In 1996 Aithal and Santhanam [1] proved that Szegö-Weineberger inequality holds true for bounded domains in noncompact rank-1 symmetric spaces, and domains contained in a geodesic ball of radius $i(M)/4$ in compact rank-1 symmetric spaces. Here and thereafter $i(M)$ denotes the injectivity radius of $M$ when $M$ is compact. Naturally, one may consider the Conjecture 1.1 in rank-1 symmetric spaces. Regarding Theorem 1.2, we consider Wang-Xia type estimate (1.4) on bounded domains in the remaining rank-1 symmetric spaces, and prove sharp isoperimetric inequalities for both compact and noncompact rank-1 symmetric spaces in this paper. Our first result is a sharp isoperimetric inequality for Neumann eigenvalues on bounded domains in compact rank-1 symmetric spaces.

Our second result is a sharp isoperimetric inequality for Neumann eigenvalues in noncompact rank-1 symmetric spaces.

Obviously, both Theorem 1.3 and Theorem 1.4 imply

recovering the main results of [1] proved by Aithal and Santhanam. Moreover if $k=1$, i.e. $M$ is a hyperbolic space of dimension $n$, $p=n-1$, hence Theorem 1.4 recovers Theorem 1.2 of [14] proved by Wang and Xia. On bounded domains in rank-1 symmetric spaces, it remains an interesting question that whether Ashbaugh and Benguria’s conjecture (Conjecture 1.1) holds or not.

The rest of the paper is organized as follows. In Section 2, we recall the geometry of rank-1 symmetric spaces. In Section 3, we give some useful tools needed in the proofs of main theorems. Sections 4 and 5 are devoted to proving Theorem 1.3 and Theorem 1.4.

Let $M$ denote any one of the following rank-1 symmetric spaces with real dimension $m$: Complex projective space $\mathbb{C}\mathbb{P}^{n}$, quarternionic projective space $\mathbb{H}\mathbb{P}^{n}$, the Cayley projective plane $\mathbf{Ca}\mathbb{P}^{2}$ or their non-compact duals. Let $\mathbb{K}$ denote $\mathbb{R}$, $\mathbb{C}$,$\mathbb{H}$ or $\mathbf{Ca}$, and $k=\dim_{\mathbb{R}}\mathbb{K}$, then $m=kn$.

We recall some basic facts about geodesic polar coordinates $(r,\xi)$ in both compact and noncompact rank-1 symmetric spaces, see [1, Section 3] and [9, Section 2] for more details. Let $o\in M$ be the center of geodesic polar coordinates, and $J(r)$ denote the Riemannian density function. Then

for $r\in(0,\pi/2)$ when $M$ is compact;

for $r>0$ when $M$ is noncompact. The trace of the second fundamental form of $S(r)$ is

namely

The Laplace operator of $M$ is given by

where $\Delta_{S_{r}}$ denotes the Laplacian of $S_{r}$. Moreover the first non-zero eigenvalue of $\Delta_{S_{r}}$ is

and the associated eigenfunctions are the linear coordinate functions restricted to $\mathbb{S}^{m-1}$, denoted by $\omega_{i}(\xi)\,(1\leq i\leq m)$, satisfying

Moreover it is known from the proof of [6, Lemma 4.11] that

where $\{\eta_{k}\}_{k=1}^{m}$ is an orthonormal basis of $T_{o}M$ with $\eta_{1}=(\omega_{1}(\xi),\cdots,\omega_{m}(\xi))$ and $\eta_{i+1}=J\eta_{i}$ for $i=1,\cdots,k-1$. Then we estimate that for compact case

for $0<r\leq\frac{\pi}{4}$, and for noncompact case

where $l$ and $p$ are defined by (1.5) and (1.8) respectively.

In this section, we will give some properties of the first nonzero eigenvalue of Neumann Laplacian and corresponding eigenfunctions on round balls, and construct trial functions for lower order Neumann eigenvalues on bounded domains in rank-1 symmetric spaces.

In this section, we consider the case of compact rank-1 symmetric spaces and prove Theorem 1.3. To begin with, we recall the following lower bound for the first nonzero Neumann eigenvalue on geodesic balls [1, Corollary 1], which will be used later on.

Now we turn to prove Theorem 1.3.

In this section, we assume $M$ is a noncompact rank-1 symmetric space. Let $\Omega$ be a bounded domain with smooth boundary in $M$, and $B$ be a geodesic ball in $M$ with volume $|\Omega|$. Denote by $R$ be the radius of $B$. The proof of Theorem 1.4 is quite similar as that of Theorem 1.3. To begin with, we first give a monotonicity lemma.

Now we turn to prove Theorem 1.4.