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Some regularity results for $p$-harmonic mappings between Riemannian manifolds (1802.01010v3)

Published 3 Feb 2018 in math.DG and math.AP

Abstract: Let $M$ be a $C2$-smooth Riemannian manifold with boundary and $N$ a complete $C2$-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 $C1$-smooth weakly $p$-harmonic mapping $u\colon M\to N$. Consequently, we obtain a Liouville-type theorem for $C1$-smooth weakly $p$-harmonic mappings in the same setting.

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