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

Chang-Yu Guo, Chang-Lin Xiang

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 $C^{1,α}$ for some $α\in (0,1)$, 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.

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

Abstract

Let be a -smooth Riemannian manifold with boundary and a complete -smooth Riemannian manifold. We show that each stationary -harmonic mapping , whose image lies in a compact subset of , is locally for some , provided that is simply connected and has non-positive sectional curvature. We also prove similar results for each minimizing -harmonic mapping with being contained in a regular geodesic ball. Moreover, when has non-negative Ricci curvature and is simply connected and has non-positive sectional curvature, we deduce a quantitative gradient estimate for each -smooth weakly -harmonic mapping . Consequently, we obtain a Liouville-type theorem for -smooth weakly -harmonic mappings in the same setting.

Paper Structure

This paper contains 11 sections, 12 theorems, 91 equations.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Main results
  4. Structure of the paper
  5. Proof of Theorem \ref{['thm:main theorem']}
  6. Proof of Theorem \ref{['thm:main theorem 2']}
  7. The $p$-minimizing tangent maps
  8. Proof of Theorem \ref{['thm:Liouville']}
  9. Gradient estimates for stationary $p$-harmonic mappings
  10. Concluding remarks
  11. $W^{2,2}$ regularity and removable singularities of $p$-harmonic mappings: $1<p<2$

Key Result

Theorem 1.1

Each stationary $p$-harmonic mapping $u\colon \Omega \to N$, whose image lies in a compact subset of $N$, is locally $C^{1,\alpha}$ for some $\alpha\in (0,1)$ if $N$ is simply connected and has non-positive sectional curvature.

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 1.2: Regular geodesic ball
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3: Monotonicity formula
  • ...and 14 more