Local Rigidity of the Bergman Metric and of the K\"ahler Carath\'eodory Metric

TL;DR

This paper proves that certain local geometric conditions on complex domains imply they are biholomorphic to a ball, establishing local rigidity results for Bergman and Carathéodory metrics.

Contribution

It introduces new local rigidity theorems linking the Kähler property of the Carathéodory metric and constant curvature of the Bergman metric to domain biholomorphism.

Findings

Carathéodory metric being locally Kähler implies the domain is a ball.

Domains with Bergman metrics of constant holomorphic sectional curvature are rigid.

Relationship between the Lu constant and local geometric properties.

Abstract

We prove that if the Carath\'eodory metric on a strictly pseudoconvex domain with a smooth boundary is locally K\"{a}hler near the boundary, then the domain is biholomorphic to a ball. We also establish a local rigidity theorem for domains with Bergman metrics of constant holomorphic sectional curvature, and highlight this relationship with the Lu constant.