On a characterization of the complex hyperbolic space

Abstract

Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ri{c}_M\ge -2(m+1).$ Assume that its universal cover has maximal bottom of spectrum ${\lambda }_1(\tilde{M}$ Then we prove that $\tilde{M}$ is isometric to the complex hyperbolic space ${\mathbb{CH}}^m.$