A Bochner formula for harmonic maps into non-positively curved metric spaces

Abstract

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(\(-\,1\)) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eells–Sampson Bochner formula in this more general setting. In particular, we show that harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density. When the domain is compact the energy density is constant, and if the domain has a point of positive Ricci curvature every harmonic map into an NPC space must be constant.