Bottom of Spectrum of Kähler Manifolds with a Strongly Pseudoconvex Boundary

Abstract

Theorem 1. Suppose (M, g) is a noncompact complete Riemannian manifold with Ric ≥ − (n− 1), we have λ0 ≤ (n− 1) /4. The estimate is sharp since the spectrum is the ray [(n− 1) /4,+∞) for the hyperbolic space H. For Kahler manifolds, this estimate can be improved. On a Kahler manifold (M, g) of complex dimension n, where g is the Riemannian metric, let ω = g (J ·, ·) be the Kahler form. In local holomorphic coordinates z1, · · · , zn, we have