A Bochner Formula for Harmonic Maps into Non-Positively Curved Metric Spaces

TL;DR

This paper extends the Bochner formula to harmonic maps from Riemannian manifolds into non-positively curved metric spaces, revealing new properties of harmonic maps related to curvature conditions.

Contribution

It develops a Bochner formula analogue for harmonic maps into CAT(-1) spaces, generalizing classical results and analyzing energy density behavior under curvature assumptions.

Findings

Harmonic maps from non-negative Ricci curvature domains have subharmonic energy density.

Energy density is constant for harmonic maps from compact domains.

Harmonic maps into NPC spaces are constant if the domain has positive Ricci curvature.

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 Eels-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.