##### Department of Mathematics,

University of California San Diego

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### Math 258

## Prof. Song-Ying Li

#### University of California, Irvine

## Bergman metric with constant curvature and uniformization theorems

##### Abstract:

This talk is based on a joint work with Xiaojun Huang entitled “Bergman metrics as pull-backs of the Fubini-Study metric”. We study domains in Cn or Stein manifolds M such that their Bergman metrics have constant holomorphic sectional curvature κ. We prove a uniformization theorem when κ < 0 through the Calabi rigidity theorem and holomorphic extension theorems. We also discuss the case when κ ≥ 0. We provide several interesting examples of the existence of such M. Under certain conditions on M, we prove that the Bergman metric of M can not have non-negative constant holomorphic sectional curvatures.

### May 11, 2023

### 1:00 PM

APM 5829

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