中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

综合报告会

（Colloquium）

报告人：Prof. Ngaiming Mok（The University of Hong Kong）

题  目：Existence, uniqueness and analytic continuation on holomorphic  isometries up to scaling with respect to the Bergman metric

时  间：09.21(星期一), 16:00-17:00 

地  点：数学院南楼N913室

Abstract:The Bergman kernel on a bounded domain  ,which is the reproducing kernel for the Hilbert space of square-integrable holomorphic functions on D, is a fundamental object in Complex Analysis. It gives rise to the Bergman metric, an Aut(D)-invariant K？hler metric serving as an intrinsic metric which has been much studied in relation to strictly pseudoconvex domains following the seminal work of C. Fefferman. We examine geometric implications for holomorphic isometries up to normalizing constants arising from regularity properties of the Bergman kernel on bounded domains which are typically weakly pseudoconvex, including especially bounded symmetric domains in their Harish-Chandra realizations. We discuss various results on existence, uniqueness and analytic continuation on holomorphic isometries up to normalizing constants between bounded domains with respect to the Bergman metric. Among other things we prove analytic continuation beyond the boundary under certain regularity assumptions on the Bergman kernel. In the case of bounded symmetric domains, we prove the existence of holomorphic isometric embeddings with respect to normalized Bergman metrics of complex unit balls of certain dimensions into irreducible bounded symmetric domains, and show that those defined on balls of maximal dimensions are unique under certain auxiliary conditions. Minimal rational curves on compact dual manifolds S of irreducible bounded symmetric domains Ω play an important role in the proofs of both existence and uniqueness, and we will make use of the geometric theory of uniruled projective manifolds established by Hwang-Mok basing on varieties of minimal rational tangents (VMRTs) and recent development in the theory of geometric structures and substructures arising from such varieties.