Infinite energy maps and rigidity

TL;DR

This paper generalizes classical rigidity results to non-compact domains, establishing the existence and properties of equivariant pluriharmonic maps into various non-compact targets, including symmetric spaces and Teichmüller spaces.

Contribution

It extends Siu's and Sampson's rigidity theorems to non-compact settings, showing existence of possibly infinite energy equivariant pluriharmonic maps and their holomorphicity under certain conditions.

Findings

Existence of $ ho$-equivariant pluriharmonic maps with possibly infinite energy.

Holomorphicity of maps when the target is Kähler and rank condition is met.

Extension of results to negatively curved Riemannian manifolds.

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.