Year: 2026
Type: preprint
Abstract: Let $Ω\subset\mathbb C^n$, $n\ge2$, 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 $K_Ω(z,z)>0$.
Source: arXiv (Cornell University)
Authors: Ebenfelt, Peter, Ganguly, Soumya
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7220677013