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

Abstract

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT(κ) spaces for κ ∈ {0, 1}. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT(κ) 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 CAT(κ) spaces. Another consequence we derive from the target variation formula is the Eells-Sampson Bochner formula for CAT(1) targets.

Quantitative gradient estimates for harmonic maps into singular spaces

“…, then the same conclusion, u is a constant map, has been proved recently by B. Freidin and Y. Zhang in [13].…”

“…Mese [35] showed that ∆e u −2κe 2 u , in the sense of distributions, for a harmonic map from a flat domain to a CAT (κ)-space. Recently, Freidin [12] and Freidin-Zhang [13] improved the method in [19] to deduce the following Bochner type inequality for a harmonic map from a Riemannian manifold into a CAT (κ)-space:…”