Upper bounds of the first eigenvalue of closed hypersurfaces by the quotient area/volume

Abstract

In this paper we obtain, for compact hypersurfaces M embedded into Hadamard manifolds, an upper sharp bound of the first closed eigenvalue. This bound depends on the isoperimetric quotient Volume(M)/Volume(Ω), where Ω is the domain enclosed by M. More precise bounds are given when the ambient space is the complex or quaternionic hyperbolic space.