通过Kähler几何、调和分析与遍历理论研究秩≥2的不可约志村簇的π₁对应的有界Γ-等变全纯映射的刚性
Rigidity of bounded $Γ$-equivariant holomorphic maps for $π_1$ of irreducible Shimura varieties of rank $\ge 2$ via Kähler geometry, harmonic analysis and ergodic theory
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中文总结 AI 辅助
本文研究秩≥2不可约志村簇的有界Γ-等变全纯映射刚性，聚焦同构定理的Carathéodory双曲靶空间变体，阐释Kähler几何、调和分析与遍历理论在证明中的核心作用。
中文摘要 AI 辅助
在作者近期的一篇论文中，我们证明了一个针对秩≥2的不可约志村簇的全纯映射的结果，称为同构定理（Isomorphism Theorem）。同构定理的证明本质上运用了Kähler几何、多复变函数论、调和分析以及遍历理论。本文我们将聚焦于同构定理的一个轻微变体，该变体中的靶空间由单连通完备Kähler-Einstein流形(M,h_M)单值化，且进一步假设该流形是Carathéodory双曲的（即由映入庞加莱圆盘的有界全纯映射空间诱导的无穷小复Finsler伪度量κ_M是一个复Finsler度量），同时Γ'⊂Aut(M)是无挠离散子群，使得商流形Y_{Γ'}:=M/Γ'关于商Kähler-Einstein度量具有有限体积。在此设定下，我们将阐释Kähler几何、调和分析与遍历理论在同构定理证明中所起的核心作用。
英文摘要
In a recent article of the authors, we proved a result called the Isomorphism Theorem for holomorphic maps from an irreducible Shimura varieties of rank $\ge 2$. The proof of the Isomorphism Theorem uses in essential ways Kähler geometry, function theory of several complex variables, harmonic analysis and ergodic theory. Here we will focus on a slight variation of the Isomorphism Theorem where the target is uniformized by a simply connected complete Kähler-Einstein manifold $(M,h_M)$ which is moreover assumed to be Carathéodory hyperbolic (i.e., the infinitesimal complex Finsler pseudometric $κ_M$ induced from the space of bounded holomorphic maps into the Poincaré disk is a complex Finsler metric) and $Γ' \subset {\rm Aut}(M)$ is a torsion-free discrete subgroup such that the quotient manifold $Y_{Γ'} := M/Γ'$ is of finite volume with respect to the quotient Kähler-Einstein metric. In this setting, we will explain the essential roles played by Kähler geometry, harmonic analysis and ergodic theory in the proof of the Isomorphism Theorem.
