Bergman度量的一个新的强刚性现象
A new strong rigidity phenomenon for the Bergman metric
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中文总结 AI 辅助
该研究建立了Bergman度量的局部到全局刚性新现象，证明两种情形下局部共形映射可延拓为双全纯映射，回答了Loi--Palmieri和Zimmer的问题，关键工具是新的Calabi型延拓定理。
中文摘要 AI 辅助
我们建立了Bergman度量的一种新的局部到全局刚性现象。也就是说，在自然的几何假设下，Bergman度量的局部共形等价可以在全局层面确定底层的复流形，仅存在移除Bergman可忽略子集这一不可避免的歧义。更确切地说，设$\Omega\subseteq\mathbb C^n$是具有完备Bergman度量的有界域，且复流形$M$的Bergman度量通过全纯映射$f$与$\Omega$的Bergman度量局部共形。我们证明，在两种互补的情形下，给定的局部映射$f$可延拓为双全纯映射$F\colon M\to D$，其像为子域$D\subseteq\Omega$。若$M$是Stein流形，则$\Omega\setminus D$是闭的多重极集。若$M$是有界域且$\Omega$满足由其自同构轨道表述的自然对称条件，则$\Omega\setminus D$是Bergman可忽略集。特别地，该结论适用于$\Omega$为有界齐性域的情形，并给出了具有局部对称Bergman度量的有界域在相差Bergman可忽略集意义下的刻画，后者回答了Loi--Palmieri和Zimmer提出的一个问题。证明的关键要素是一个专为Bergman度量定制的新的Calabi型延拓定理。
英文摘要
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 $Ω\subseteq\mathbb C^n$ 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 $Ω$. We prove that the given local map $f$ extends to a biholomorphism $F\colon M\to D$ onto a subdomain $D\subseteqΩ$ in two complementary settings. If $M$ is Stein, then $Ω\setminus D$ 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 $Ω\setminus D$ is Bergman-negligible. In particular, this applies when $Ω$ is a bounded homogeneous domain and yields a characterization, up to Bergman-negligible sets, of bounded domains with locally symmetric Bergman metrics. The latter answers a question raised by Loi--Palmieri and Zimmer. A key ingredient in the proof is a new Calabi-type extension theorem tailored to Bergman metrics.
