Infinite energy maps and rigidity

Abstract

We extend Siu's and Sampson's celebrated rigidity results to non-compact domains. More precisely, let $M$ be a smooth quasi-projective variety with universal cover $\tilde M$ and let $\tilde X$ be a symmetric space of non-compact type, a locally finite Euclidean building or the Weil-Petersson completion of the Teichm\"uller space of a surface of genus $g$ and $p$ punctures with $3g-3+p>0$. Under suitable assumptions on a homomorphism $\rho: \pi_1(M) \rightarrow \mathsf{Isom}(\tilde X)$, we show that there exists a $\rho$-equivariant pluriharmonic map $\tilde u: \tilde M \rightarrow \tilde X$ of possibly infinite energy. In the case when the target is K\"ahler and $\mathsf{rank}(d \tilde u) \geq 3$ at some point, $\tilde u$ is holomorphic or conjugate holomorphic. This builds on previous important work by Jost-Zuo and Mochizuki. We also extend these results to the case when the target is a Riemannian manifold with sectional curvature bounded from above by a negative constant.