On the Bergman metric of symmetric spaces

TL;DR

This paper investigates when the Bergman metric on bounded domains in complex space is locally symmetric, establishing conditions under which the domain is globally symmetric or biholomorphic to a symmetric domain minus a pluripolar set.

Contribution

It proves new rigidity results linking local symmetry of the Bergman metric to global symmetry or biholomorphic equivalence to symmetric domains, using advanced geometric and analytic methods.

Findings

Complete Bergman metric implies the domain is globally symmetric.

Pseudoconvex domains with locally symmetric Bergman metric are biholomorphic to symmetric domains minus pluripolar sets.

The results unify structure theory of symmetric spaces with complex analysis techniques.

Abstract

We study bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in \cite{UnifThm2}, we obtain two rigidity results. If the Bergman metric of $\Omega$ is complete, then $\Omega$ is (globally) symmetric. If instead $\Omega$ is pseudoconvex, then $\Omega$ is biholomorphic to $\widetilde\Omega\setminus E$, where $\widetilde\Omega\subset\mathbb{C}^n$ is a bounded symmetric domain and $E\subset\widetilde\Omega$ is relatively closed and pluripolar. The proofs combine the structure theory of Hermitian symmetric spaces with Calabi's theory of K\"ahler immersions into the infinite dimensional complex projective space (in particular, rigidity and the hereditary property of the diastasis), together with analytic and pluripotential tools based on extension properties of square-integrable holomorphic functions and the Bergman kernel.