Local Rigidity of the Bergman Metric and of the Kähler Carathéodory Metric

Abstract

We prove that if the Carathéodory metric on a strictly pseudoconvex domain with a smooth boundary is locally Kähler 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.

19 pages, final version to appear in the Bulletin of the London Mathematical Society