On a characterization of the complex hyperbolic space

TL;DR

This paper characterizes complex hyperbolic space by showing that a compact Kähler manifold with a Ricci curvature lower bound and a universal cover with maximal spectral bottom must be isometric to complex hyperbolic space.

Contribution

It provides a new characterization of complex hyperbolic space based on spectral and curvature conditions, extending previous geometric rigidity results.

Findings

Universal cover is isometric to complex hyperbolic space under given conditions.

Spectral bottom achieves maximum value for the universal cover.

Ricci curvature lower bound is crucial for the characterization.

Abstract

Consider a compact K\"{a}hler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $% \widetilde{M}$ has maximal bottom of spectrum $\lambda_1(\widetilde{M}%) =m^2.$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$