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

TL;DR

The paper characterizes when canonical invariant metrics on bounded pseudoconvex domains reveal a ball as the underlying domain. It proves that a Carathéodory metric that is locally Kähler near the boundary on a smoothly bounded strictly pseudoconvex domain forces the domain to be biholomorphic to the ball, and it establishes a local-to-global rigidity principle for Bergman metrics: constant negative holomorphic sectional curvature on an open set yields a global isometry into a ball via Bergman representative coordinates, leading to Lu-constant–driven rigidity results. The work unifies local curvature conditions with global domain structure, strengthens Lu's theorem by requiring curvature constancy only locally, and derives corollaries for domains with spherical boundaries and equality cases in Lu's inequality. These results provide new geometric criteria for identifying ball domains through metric properties and enhance understanding of the interplay between Carathéodory, Bergman, and Kobayashi metrics in several complex variables.

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.

Theorems & Definitions (24)

• Theorem 1.1: W
• Theorem 1.2: Sta
• Theorem 1.3: GZ
• Theorem 1.4
• Theorem 1.5: Look, cf. CXZ
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8: GZ
• Theorem 1.9
• Theorem 1.10