Comparison theorem for Kähler manifolds and positivity of spectrum (Q814678)

The authors consider an \(m\)-dimensional complete Kähler manifold \(M\), with holomorphic bisectional curvature bounded from below by \(-1\); they prove that the bottom of the spectrum has an upper bound given by \(\lambda_1(M) \leq m^2\). If moreover \(\lambda_1(m) \geq m\), then \(M\) must have only one end with infinite volume. A complete Kähler surface with holomorphic bisectional curvature bounded from below by \(-1\) and satisfying \(\lambda_1(M) \geq 4\) must have at most 4 ends (three with finite volume and the fourth with infinite volume).