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 Teichmüller space of a surface of genus $g$ and $p$ punctures with $3g-3+p>0$. Under suitable assumptions on a homomorphism $\rho :{\pi }_1(M)\to \mathsf{Isom}(\tilde{X})$, we show that there exists a $\rho$-equivariant pluriharmonic map $\tilde{u}:\tilde{M}\to \tilde{X}$ of possibly infinite energy. In the case when the target is Kähler and $\mathsf{rank}(d\tilde{u})\ge 3$ 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.

Cited by in corpus (1)

• Pluriharmonic maps into buildings and symmetric differentials