Communications in Analysis and Geometry

Volume 18 (2010)

Number 2

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

Pages: 375 – 395

DOI: http://dx.doi.org/10.4310/CAG.2010.v18.n2.a5

Authors

Song-Ying Li (Department of Mathematics, University of California at Irvine)

My-An Tran (Department of Mathematics, University of California at Irvine)

Abstract

The research in paper is a continuation of the work of Li and Wang \cite{LW1,LW2, LW3} who studied upper estimates for $\lambda_1=\lambda_1(\Delta_g)$, thebottom of the spectrum of Laplace–Beltrami operator on a complete non-compactKähler manifold $(M^n, g)$ with a lower bound condition on holomorphicbisectional curvature and the work of Munteanu \cite{M} who uses lower boundcondition on Ricci curvature. In this paper, we study the problems on a boundedpseudoconvex domain $D$ in $\mathbb{C}^n$ with a certain normalized completeKähler metrics $u$ on $D$ which are called Bergman-type, we find a class ofBergman-type metrics $u$ on $D$ so that $\lambda_1(\Delta_u)=n^2$. We alsoprovide a simple condition on metric $u$, under this condition, we obtain thesharp upper bound estimates $n^2$ for $\lambda_1(\Delta_u)$ for such class ofBergman-type metrics, which include Kähler–Einstein metric and Bergman metricon $D$.

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