发表机构
University of California San Diego; Rutgers University–New Brunswick（加州大学圣地亚哥分校; 罗格斯大学新不伦瑞克校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有界伪凸域上Bergman度量的极值性、常数量曲率与Kähler--Einstein条件等价，并推广到无界情形，核心工具是ACH边界处的唯一延拓定理。
AI 中文摘要
设 $\Omega\subset\mathbb C^n$，$n\ge2$，为有界连通伪凸域，其边界包含一个光滑强伪凸点。我们证明：若 $\Omega$ 的Bergman度量是极值的，则其数量曲率恒等于 $-n$；并且常数量曲率迫使Bergman度量成为Kähler--Einstein度量。因此，在此情形下，极值性、常数量曲率与Kähler--Einstein条件三者等价。一个关键的分析工具是在ACH边界处关于Bergman Laplace算子的局部唯一延拓定理，该定理利用Hörmander的Carleman估计证明。我们还得到了对可能无界伪凸域的推广：若Bergman度量在该类边界点附近是cscK或极值的，则它在 $K_\Omega(z,z)>0$ 的整个区域上良定义且为Kähler--Einstein度量。
英文摘要
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 $K_Ω(z,z)>0$.
