Infimum of the spectrum of Laplace-Beltrami operator on a bounded pseudoconvex domain with a Kähler metric of Bergman type

where dVg is the volume measure on M with respect to the Kähler metric g. When M is compact and ∆g is uniformly elliptic, λ1(∆g) is the first positive eigenvalue of ∆g with Dirichlet boundary condition. A lot of research has been done on its upper and lower bound estimates and its impact on geometry and physics (see for examples, the lecture notes of P. Li [8] and the paper of S. Udagawa [16] and references therein). When (M, g) is a complete noncompact Kähler manifold, λ1(∆g) may not be an eigenvalue of ∆g. For example, when M is the complex hyperbolic space CH, λ1(∆g) is no longer an L 2 eigenvalue of ∆g. However, it is the