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arXiv · 2610.04590

Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains

Peter Ebenfelt·Soumya Ganguly
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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 $K_Ω(z,z)>0$.

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Peter Ebenfelt, Soumya Ganguly. 2026-10-03. Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains. https://arxiv.org/abs/2610.04590

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