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Boundary partial regularity for a class of Dirac-harmonic maps

Jürgen Jost, Jingyong Zhu

TL;DR

The paper proves boundary partial regularity for coupled Dirac-harmonic maps under a near-boundary energy monotonicity inequality, establishing Hölder continuity away from a small singular set and, with smoother boundary data and larger spinor integrability, $C^{1,\mu}$ regularity. The approach combines Hélein's adapted frame construction with Morrey-type decay estimates and relates boundary behavior to interior regularity results. A key contribution is extending boundary regularity theory to a coupled system where the map and spinor interact through curvature and the Dirac operator, under precise energy-growth control. The results advance understanding of boundary behavior for geometrically natural variational problems arising from supersymmetric sigma models and have implications for higher-regularity theories in coupled elliptic systems.

Abstract

In this paper, we prove the boundary partial regularity for a class of coupled Dirac-harmonic maps satisfying a certain energy monotonicity inequality near the boundary.

Boundary partial regularity for a class of Dirac-harmonic maps

TL;DR

The paper proves boundary partial regularity for coupled Dirac-harmonic maps under a near-boundary energy monotonicity inequality, establishing Hölder continuity away from a small singular set and, with smoother boundary data and larger spinor integrability, regularity. The approach combines Hélein's adapted frame construction with Morrey-type decay estimates and relates boundary behavior to interior regularity results. A key contribution is extending boundary regularity theory to a coupled system where the map and spinor interact through curvature and the Dirac operator, under precise energy-growth control. The results advance understanding of boundary behavior for geometrically natural variational problems arising from supersymmetric sigma models and have implications for higher-regularity theories in coupled elliptic systems.

Abstract

In this paper, we prove the boundary partial regularity for a class of coupled Dirac-harmonic maps satisfying a certain energy monotonicity inequality near the boundary.

Paper Structure

This paper contains 4 sections, 5 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Small energy boundary regularity
  4. Proof of Theorem \ref{['holder']}

Key Result

Theorem 1.1

Let $(u,\psi)$ be a coupled weakly Dirac-harmonic map from a compact manifold $M$ of dimension $m\geq2$ into another compact Riemannian manifold $N$ of dimension $n\geq2$, and $u=\phi\in C^{0,1}(\partial M,N)$ in the sense of trace. Suppose $\psi\in L^{q}(M)$ for some $q>2m$ and $u$ satisfies the en for all $x\in M_{\rho_0}:=\{x\in\bar{M}: {\rm dist}(x,\partial M)< \rho_0\}$ and $0<s\leq r<\rho_0$

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Lemma \ref{['M']}
  • ...and 2 more