The upper bound of the harmonic mean of the Neumann eigenvalues in curved spaces

Abstract

Considering an n-dimensional Riemannian manifold M whose sectional curvature is bounded above by κ and Ricci curvature is bounded below by (n−1)K, we obtain an upper bound for the harmonic mean of the first (n−1) non-zero Neumann eigenvalues of the Laplacian for domains contained in M. This can be viewed as certain isoperimetric inequality and generalizes the results for domains in space forms [5,15].