Heat flow of p-harmonic maps from complete manifolds into generalised regular balls

Abstract

We study the heat flow of p-harmonic maps between complete Riemannian manifolds. We prove the global existence of the flow when the initial datum has values in a generalised regular ball. In particular, if the target manifold has nonpositive sectional curvature, we obtain the global existence of the flow for any initial datum with finite p-energy. If, in addition, the target manifold is compact, the flow converges to a p-harmonic map. This gives an extension of the results of Liao-Tam [12] concerning the harmonic heat flow (p = 2) to the case p $\ge$ 2. We also derive a Liouville type theorem for p-harmonic maps between complete Riemannian manifolds.