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

Christine Breiner, Ailana Fraser, Lan-Hsuan Huang, Chikako Mese, Pam Sargent, Yingying Zhang

Published 2016-10-25Version 1

We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove H\"older regularity with H\"older constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $(n-2)$-skeleton, we improve the regularity to locally Lipschitz. Finally, for points $x \in X^{(k)}$ with $k \leq n-2$, we demonstrate that the H\"older exponent depends on geometric and combinatorial data of the link of $x \in X$.