An upper bound for the first nonzero Neumann eigenvalue

Abstract

Let M denote a complete, simply connected n-dimensional Riemannian manifold with sectional curvature KM≤k and Ricci curvature RicM≥(n−1)K, where k,K∈R. Then for a bounded domain Ω⊂M with smooth boundary, we prove that the first nonzero Neumann eigenvalue μ1(Ω)≤Cμ1(Bk(R)). Here Bk(R) is a geodesic ball of radius R>0 in the simply connected space form Mk such that vol(Ω) = vol(Bk(R)), and C is a constant which depends on the volume, diameter of Ω and the dimension of M.