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# Communications in Analysis and Geometry

## Volume 28 (2020)

### Number 4

### The First of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

### Existence of harmonic maps into CAT(1) spaces

Pages: 781 – 835

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n4.a2

#### Authors

#### 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 nontrivial 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.

C. Breiner and C. Mese were supported in part by NSF grants DMS-1308420 and DMS-1406332 respectively.

L.-H. Huang was supported by NSF grants DMS-1308837 and DMS-1452477.

A. Fraser was supported in part by an NSERC Discovery Grant.

P. Sargent was supported in part by an NSERC PGS D scholarship and a UBC Four Year Doctoral Fellowship.

Y. Zhang was supported in part by an AWM-NSF Travel Grant.

This material is also based upon work supported by NSF DMS-1440140 while C.B. and A.F. were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester.

Received 14 January 2018

Accepted 15 January 2018

Published 1 October 2020