Existence of harmonic maps into CAT(1) spaces

Abstract

Let $\phi \in C^0\cap W^{1,2}(\Sigma ,X)$ 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 $\phi$ 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.

References in corpus (1)

• Regularity of Harmonic Maps from Polyhedra to CAT(1) Spaces