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

TL;DR

This paper investigates the heat flow of p-harmonic maps between complete Riemannian manifolds, establishing global existence and convergence results under certain curvature and energy conditions, extending previous harmonic map results to p-harmonic cases.

Contribution

It extends the theory of harmonic heat flow to p-harmonic maps for p ≥ 2, proving global existence and convergence results in broader geometric settings.

Findings

Global existence of p-harmonic heat flow with initial data in regular balls.

Flow converges to p-harmonic maps when the target is compact.

Liouville type theorem for p-harmonic maps between complete manifolds.

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.