Sharp upper bound for the first eigenvalue

Abstract

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