Rigidity of bounded $\Gamma$-equivariant holomorphic maps for $\pi_1$ of irreducible Shimura varieties of rank $\ge 2$ via K\"ahler 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 K\"ahler 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 K\"ahler-Einstein manifold $(M,h_M)$ which is moreover assumed to be Carath\'eodory hyperbolic (i.e., the infinitesimal complex Finsler pseudometric $\kappa_M$ induced from the space of bounded holomorphic maps into the Poincar\'e disk is a complex Finsler metric) and $\Gamma' \subset {\rm Aut}(M)$ is a torsion-free discrete subgroup such that the quotient manifold $Y_{\Gamma'} := M/\Gamma'$ is of finite volume with respect to the quotient K\"ahler-Einstein metric. In this setting, we will explain the essential roles played by K\"ahler geometry, harmonic analysis and ergodic theory in the proof of the Isomorphism Theorem.