On the Bergman metric of symmetric spaces

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.