Rigidity of bounded $\Gamma$-equivariant holomorphic maps for ${\pi }_1$ of irreducible Shimura varieties of rank $\ge 2$ via Kähler geometry, harmonic analysis and ergodic theory

Abstract

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 ${\kappa }_M$ induced from the space of bounded holomorphic maps into the Poincaré disk is a complex Finsler metric) and ${\Gamma }^{\prime }\subset \mathrm{Aut}(M)$ is a torsion-free discrete subgroup such that the quotient manifold $Y_{{\Gamma }^{\prime }}:=M/{\Gamma }^{\prime }$ 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.

Proceedings of the International Consortium of Chinese Mathematicians