Gradient estimate and Liouville theorems for p-harmonic maps

TL;DR

This paper establishes gradient estimates and Liouville theorems for p-harmonic maps, providing new insights into their geometric properties and applications under curvature constraints.

Contribution

It introduces an $L^q$ gradient estimate for p-harmonic maps and derives Liouville theorems, extending understanding of their behavior on manifolds with curvature bounds.

Findings

Derived $L^q$ gradient estimates for p-harmonic maps.

Established Liouville theorems for p-harmonic maps.

Applied results to geometric problems involving curvature conditions.

Abstract

In this paper, we first obtain an $L^q$ gradient estimate for $p$-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this $L^q$ gradient estimate, we get a corresponding Liouville type result for $p$-harmonic maps. Secondly, using these general results, we give various geometric applications to $p$-harmonic maps from complete manifolds with nonnegative Ricci curvature to manifolds with various upper bound on sectional curvature, under appropriate controlled images.