We study the roles of domain and target curvatures in harmonic maps into metric spaces with upper curvature bounds. We begin computing the domain and target variation formulas, incorporating the roles of the domain Ricci and scalar curvatures in the former, and the target's curvature bound in the latter. These formulas are optimal in the sense that they match the classical formulas in the case when the target is a smooth manifold. We then use these formulas to derive a generalization of the Eells-Sampson Bochner formula for this case of singular targets, generalizing the work of Chen who proved a subharmonicity statement under stronger geometric assumptions. Our formula includes terms involving domain Ricci and target curvature terms that are analogous to terms from the Eells-Sampson formula. As a consequence of this formula we derive rigidity results for harmonic maps, including reproving a result of Korevaar and Schoen on rigidity of harmonic maps from flat tori to non-positively curved spaces. We derive more consequences from our domain and target variation formulas. First, we conclude that the distance squared of the map to a point is a weakly subharmonic function if the image is convex, and use this, along with a reverse Poincar\'e inequality, to prove a Liouville-type theorem for harmonic maps from Riemannian polyhedra to spaces with positive curvature bounds. We also analyze conformal harmonic maps to singular spaces; for domains of dimension at least $3$, we generalize a result from the theory of harmonic morphisms, that such maps are rescaled isometric immersions. In dimension $2$ we generalize a well known result that gives an a-priori energy bound for conformal harmonic maps from hyperbolic surfaces to spaces with curvature bounded above by a negative constant.
Freidin, Brian,
"Harmonic maps between and into singular spaces"
(2018).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/qahz-hp74
