Some regularity results for $p$-harmonic mappings between Riemannian manifolds

TL;DR

This paper establishes regularity and gradient estimates for p-harmonic mappings between Riemannian manifolds under curvature conditions, extending understanding of their smoothness and behavior.

Contribution

It proves local $C^{1,eta}$ regularity and gradient bounds for p-harmonic maps under specific geometric conditions, including new Liouville-type theorems.

Findings

Stationary p-harmonic maps are locally $C^{1,eta}$ under curvature assumptions.

Gradient estimates are derived for weakly p-harmonic maps with non-negative Ricci curvature.

Liouville-type theorems are established for certain p-harmonic mappings.

Abstract

Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each stationary $p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of $N$, is locally $C^{1,\alpha}$ for some $\alpha\in (0,1)$, provided that $N$ is simply connected and has non-positive sectional curvature. We also prove similar results for each minimizing $p$-harmonic mapping $u\colon M\to N$ with $u(M)$ being contained in a regular geodesic ball. Moreover, when $M$ has non-negative Ricci curvature and $N$ is simply connected and has non-positive sectional curvature, we deduce a quantitative gradient estimate for each $C^1$-smooth weakly $p$-harmonic mapping $u\colon M\to N$. Consequently, we obtain a Liouville-type theorem for $C^1$-smooth weakly $p$-harmonic mappings in the same setting.