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Deforming a map into a harmonic map

Let X be a complete noncompact Riemannian manifold with Ricci curvature and Sobolev radius (see §6 for the definition) bounded from below and Y a complete Riemannian manifold with nonpositive sectional curvature. We shall study some situations where a smooth map f : X → Y can be deformed continuously into a harmonic map, using a naturally defined flow. The flow used here is not the usual harmonic heat flow, as introduced by Eells–Sampson. We use, instead, a flow introduced by J.P. Anderson [1]. Except for some classical results on linear elliptic partial differential equations, this paper is self–contained and provides a straightforward proof for a wide range of existence and uniqueness theorems for harmonic maps. In particular, we obtain as a corollary a recent result of Hardt–Wolf [7] on the existence of harmonic quasiisometries of the hyperbolic plane.