Comparison theorem for Kähler manifolds and positivity of spectrum

Abstract

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 ℙℂ^m, ℙ^m, and ℙℍ^m. 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.