Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second 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

A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces

Pages: 1847 – 1862



Brian Freidin (Department of Mathematics, Vancouver, British Columbia, Canada)

Yingying Zhang (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We study analytic properties of harmonic maps from Riemannian polyhedra into $\operatorname{CAT}(\kappa)$ spaces for $\kappa \in {\lbrace 0, 1 \rbrace}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into $\operatorname{CAT}(\kappa)$ spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an $L^p$ Liouville-type theorem for harmonic maps from admissible polyhedra into convex $\operatorname{CAT}(\kappa)$ spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for $\operatorname{CAT}(1)$ targets.

Received 2 April 2018

Accepted 20 November 2019

Published 8 January 2021