Existence of harmonic maps into CAT(1) spaces

TL;DR

This paper proves the existence of harmonic maps from Riemann surfaces into compact CAT(1) spaces, using harmonic replacement techniques and establishing compactness and singularity removal results.

Contribution

It introduces a method to establish harmonic map existence into CAT(1) spaces, including compactness and singularity theorems, extending harmonic map theory to non-smooth targets.

Findings

Existence of harmonic maps into CAT(1) spaces under certain conditions

Compactness of energy minimizers in this setting

Removable singularity theorem for conformal harmonic maps

Abstract

Let $\varphi\in C^0 \cap W^{1,2}(\Sigma, X)$ where $\Sigma$ is a compact Riemann surface, $X$ is a compact locally CAT(1) space, and $W^{1,2}(\Sigma,X)$ is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map $u:\Sigma \to X$ homotopic to $\varphi$ or there exists a conformal harmonic map $v:\mathbb S^2 \to X$. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.