  One of the main purposes of this paper is to prove that on a complete
K\"ahler 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 $\Bbb R \times N$,
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 K\"ahler manifolds satisfying a weighted Poincar\'e
inequality are also being considered
