Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains
Abstract
Let Ω⊂ℂn, n≥2, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of Ω is extremal, then its scalar curvature is identically −n, and that constant scalar curvature forces the Bergman metric to be Kähler--Einstein. Consequently, extremality, constant scalar curvature, and the Kähler--Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using Hörmander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and Kähler--Einstein throughout the locus where KΩ(z,z)>0.
Related
- CSC Bergman metrics are Einstein on pseudoconvex domains
- Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces
- Comparison of the Bergman and Szegő kernels on arbitrary bounded pseudoconvex domains with C2 boundary
- Strongly Pseudoconvex Domains with Finite Dimensional p-Bergman Spaces
- Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
BibTeX
@misc{ebenfelt2026rigidity,
title = {{Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains}},
author = {Peter Ebenfelt and Soumya Ganguly},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04590},
eprint = {2610.04590},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04590},
}
RIS
TY - UNPB
TI - Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains
AU - Peter Ebenfelt
AU - Soumya Ganguly
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04590
UR - https://arxiv.org/abs/2610.04590
AB - Let Ω⊂ℂ^n, n≥2, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of Ω is extremal, then its scalar curvature is identically −n, and that constant scalar curvature forces the Bergman metric to be Kähler--Einstein. Consequently, extremality, constant scalar curvature, and the Kähler--Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using Hörmander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and Kähler--Einstein throughout the locus where K_Ω(z,z)>0.
ER -