Sharp upper bound for the first eigenvalue

Abstract

Let \(M\) be a closed hypersurface in a noncompact rank-1 symmetric space \((\overline{\mathbb{M }}, ds^2)\) with \(-4 \le K_{\overline{\mathbb{M }}} \le -1,\) 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 \(M\).