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

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 ${\mathrm{\Gamma }}^{\mathrm{\prime }}\subset \mathrm{A}\mathrm{u}\mathrm{t}(M)$ is a torsion-free discrete subgroup such that the quotient manifold $Y_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}:=M\mathrm{/}{\mathrm{\Gamma }}^{\mathrm{\prime }}$ 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.