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 CP, C, and CH. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m 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. 0. Introduction In 1975, Cheng [1] proved a comparison theorem for the first Dirichlet eigenvalues of the Laplacian on geodesic balls. One of the consequences is a sharp upper bound for the bottom of the spectrum on a complete manifold with Ricci curvature bounded from below. Theorem 0.1 (Cheng). Let Mn be a complete Riemannian manifold of dimension n. Suppose the Ricci curvature of M has a lower bound given by RicM ≥ −(n− 1). Then, the bottom of the spectrum of the Laplacian must satisfy the upper bound λ1(M) ≤ (n− 1) 2