A new strong rigidity phenomenon for the Bergman metric

Abstract

We establish a new local-to-global rigidity phenomenon for the Bergman metric. Namely, under natural geometric hypotheses, a local conformal identification of Bergman metrics determines the underlying complex manifold globally, up to the unavoidable ambiguity of removing Bergman-negligible subsets. More precisely, let $Ω\subseteq\mathbb C^n$ be a bounded domain with a complete Bergman metric, and suppose that the Bergman metric of a complex manifold $M$ is locally conformal, via a holomorphic map $f$, to that of $Ω$. We prove that the given local map $f$ extends to a biholomorphism $F\colon M\to D$ onto a subdomain $D\subseteqΩ$ in two complementary settings. If $M$ is Stein, then $Ω\setminus D$ is a closed pluripolar set. If $M$ is a bounded domain and $Ω$ satisfies a natural symmetry condition expressed in terms of its automorphism orbits, then $Ω\setminus D$ is Bergman-negligible. In particular, this applies when $Ω$ is a bounded homogeneous domain and yields a characterization, up to Bergman-negligible sets, of bounded domains with locally symmetric Bergman metrics. The latter answers a question raised by Loi--Palmieri and Zimmer. A key ingredient in the proof is a new Calabi-type extension theorem tailored to Bergman metrics.