Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces

Abstract

In this paper, we prove that for a bounded domain Ω \Omega in a rank- 1 1 symmetric space, the first non-zero Neumann eigenvalue μ 1 ( Ω ) ≤ μ 1 ( B ( r 1 ) ) \mu _{1}(\Omega )\leq \mu _{1}(B(r_{1})) where B ( r 1 ) B(r_{1}) denotes the geodesic ball of radius r 1 r_{1} such that v o l ( Ω ) = v o l ( B ( r 1 ) ) \begin{equation*}vol(\Omega )=vol(B(r_{1}))\end{equation*} and equality holds iff Ω = B ( r 1 ) \Omega =B(r_{1}) . This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.