Infimum of the spectrum of Laplace Beltramioperator on a bounded pseudoconvex domain witha Kahler metric of Bergman type

摘要

The research in paper is a continuation of the work of Li and Wang[10-12] who studied upper estimates for λ_1 = λ_1(△ _g)the bottom of the spectrum of Laplace—Beltrami operator on a complete non-compact Kahler manifold (М~n, g) with a lower bound condition on holomorphic bisectional curvature and the work of Munteanu [16] who uses lower bound condition on Ricci curvature. In this paper, we study the problems on a bounded pseudoconvex domain D in (C~n with a certain normalized complete Kahler metrics u on D which are called Bergman-type, we find a class of Bergman-type metrics a on D so that λ_1 (△_u) = n~2. We also provide a simple condition on metric u, under this condition, we obtain the sharp upper bound estimates n~2 for λ_1 (△_u) for such class of Bergman-type metrics, which include Kahler—Einstein metric and Bergman metric on D.