Sharp upper bound for the first eigenvalue

Abstract

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}},d{s}^2)$ with $-4\le K_{\bar{\mathbb{M}}}\le -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0\le K_{\mathbb{M}}\le {\delta }^2$ or $K_{\mathbb{M}}\le k$ where $k=-{\delta }^2$ or 0. In this paper we give sharp upperbounds for the first eigenvalue of laplacian of $M$.