Connectedness at infinity of complete Kähler manifolds

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 isometric to ${\Bbb R} \times N$ with a specialized metric, with $N$ being compact. Generalizations to complete K\"ahler manifolds satisfying a weighted Poincar\'e inequality are also being considered.