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

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 $u:M\to N$, 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 $u:M\to N$ 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 $u:M\to N$. Consequently, we obtain a Liouville-type theorem for $C^1$-smooth weakly $p$-harmonic mappings in the same setting.

23 pages; we made several changes according to the suggestions of the referees

References in corpus (1)

• Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps