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

摘要

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