Deforming a map into a harmonic map

TL;DR

This paper uses a flow method to establish existence theorems for harmonic maps from noncompact Riemannian manifolds, including extensions of quasisymmetric sphere maps to harmonic diffeomorphisms of hyperbolic space.

Contribution

It provides a simpler proof of existence theorems for harmonic maps and extends results on quasisymmetric sphere maps to harmonic diffeomorphisms of hyperbolic space.

Findings

Existence theorems for harmonic maps from noncompact manifolds.

Extension of quasisymmetric sphere maps to harmonic diffeomorphisms.

Simplified proof technique for harmonic map existence.

Abstract

Using a flow first introduced by J.P. Anderson, we obtain some existence theorems for harmonic maps from a noncompact complete Riemannian manifold into a complete Riemannian manifold. In particular, we prove as a corollary a recent result of Hardt and Wolf stating that any quasisymmetric map of the sphere that is sufficiently close to the identity can be extended to a quasiconformal harmonic diffeomorphism of the hyperbolic ball. This version contains a much simpler proof than the first version.