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

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT($\kappa$) spaces for $\kappa \in \{0,1\}$. 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.