Comparison theorem for Kähler manifolds and positivity of spectrum

The first part of this paper is devoted to proving a comparison theorem for Kähler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are ℙℂm, ℙm, and ℙℍm. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m2 for a complete, m-dimensional, Kähler manifold with holomorphic bisectional curvature bounded from below by −1. The second part of the paper is to show that if this upper bound is achieved and when m=2, then it must have at most four ends.