Bounded holomorphic functions on open Kaehler manifolds with negative sectional curvature
主 题: Bounded holomorphic functions on open Kaehler manifolds with negative sectional curvature
报告人: Prof. Jianguo Cao (Notre Dame University)
时 间: 2011-05-25 14：00-15：00
地 点: 理科一号楼1114（数学所活动）
In this lecture, we discuss an open problem of Siu-Yau-Wu on Kaehler manifolds with negative sectional curvature. Suppose that M^n is a complete and simply-connected Kaehler manifold with negative sectional curvature between two constant numbers -b and -1. It has been conjectured by various authors that such a complex manifold admits a non-trivial bounded holomorphic function. Using the results in comparison geometry, we first construct a non-trivial bounded strictly pluri-sub-harmonic function f with respect to a new metric g on such a complex manifold M^n. Afterwards, we study Kohn's Laplace operators on the level CR-hypersurfaces of the function f. When complex dimension n of M^n is great than or equal to 3, we apply the earlier Holder estimates of Lihe Wang and Mei-Chi Shaw to the Kohn's Laplace operators, in order to construct a family of bounded non-trivial CR-functions on the level CR-hypersurfaces of f. By the Kohn-Rossi theory, such a family of uniform bounded CR-functions would produce a desired non-trivial bounded holomorphic function on M^n, when n >= 3. This is an ongoing joint work with Mei-Chi Shaw. This lecture is accessible to advanced graduate students and non-experts.