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⊆ℂn 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, with a constant conformal factor when n=1. We prove that the given local map f extends to a biholomorphism F: M→ D onto a subdomain D⊆Ω in two complementary settings. If M is Stein, then Ω∖ 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∖ 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 a class of bounded domains with locally symmetric Bergman metrics, addressing a question raised by Loi--Palmieri and Zimmer. A key ingredient in the proofs is a new Calabi-type extension theorem tailored to Bergman metrics.
Related
- Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
- Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains
- Rigidity of positive mass theorem with fast metric decay
- Kähler-Finsler rigidity of the Kobayashi metric and the Kähler base of weakly Kähler-Finsler metrics
- The anisotropic Calderón problem: rigidity near the Euclidean metric
BibTeX
@misc{ebenfelt2026strong,
title = {{A new strong rigidity phenomenon for the Bergman metric}},
author = {Peter Ebenfelt and John N. Treuer and Ming Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15998},
eprint = {2608.15998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15998},
}
RIS
TY - UNPB
TI - A new strong rigidity phenomenon for the Bergman metric
AU - Peter Ebenfelt
AU - John N. Treuer
AU - Ming Xiao
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2608.15998
UR - https://arxiv.org/abs/2608.15998
AB - 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⊆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, with a constant conformal factor when n=1. We prove that the given local map f extends to a biholomorphism F: M→ D onto a subdomain D⊆Ω in two complementary settings. If M is Stein, then Ω∖ 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∖ 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 a class of bounded domains with locally symmetric Bergman metrics, addressing a question raised by Loi--Palmieri and Zimmer. A key ingredient in the proofs is a new Calabi-type extension theorem tailored to Bergman metrics.
ER -