Connectedness at infinity of complete Kähler manifolds

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 λ ₁(M) ≥ m ², then it must either be connected at infinity or isometric to ℝ×N with a specialized metric, with N being compact. Generalizations to complete Kähler manifolds satisfying a weighted Poincar'e inequality are also being considered.