Connectedness at infinity of complete Kähler manifolds and locally symmetric spaces

Abstract

One of the main purposes of this paper is to prove that on a complete Kähler manifold of dimension $m$, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum ${\lambda }_1(M)\ge m^2$, then it must either be connected at infinity or diffeomorphic to $\mathbb{R}\times N$, where $N$ is a compact quotient of the Heisenberg group. Similar type results are also proven for irreducible, locally symmetric spaces of noncompact type. Generalizations to complete Kähler manifolds satisfying a weighted Poincaré inequality are also being considered