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Coulomb Gauges and Regularity for Stationary Weak Yang$-$Mills Connections in Supercritical Dimension

Riccardo Caniato, Tristan Rivière

TL;DR

This work addresses regularity for stationary weak Yang–Mills connections in the supercritical dimension $n=5$ within the weak-connection framework. It develops a Morrey-norm based Coulomb-gauge extraction and a two-stage approximation scheme that smoothe weak connections into smooth ones while preserving controlled Morrey norms of the curvature. The authors prove an epsilon-regularity theorem and deduce partial regularity, establishing that the singular set has vanishing $1$-dimensional Hausdorff measure, thereby extending prior results for smooth, Sobolev, and admissible connections to the broader weak-connection setting. The methods rely on extension and gluing techniques on cubes, admissible covers, and careful trace control, significantly advancing the understanding of Yang–Mills fields in dimensions above the critical threshold and enabling robust bubbling and energy-quantization analyses.

Abstract

We prove that stationary Yang$-$Mills fields in dimensions 5 belonging to the variational class of weak connections are smooth away from a closed singular set $S$ of vanishing 1-dimensional Hausdorff measure. Our proof is based on an $\varepsilon$-regularity theorem, which generalizes to this class of weak connections the existing previous $\varepsilon$-regularity results by G. Tian for smooth connections, by Y. Meyer and the second author for Sobolev and approximable connections, and by T. Tao and G. Tian for admissible connections (which are weak limits of smooth Yang$-$Mills fields). On the path towards establishing $\varepsilon$-regularity, a pivotal step is the construction of controlled Coulomb gauges for general weak connections under small Morrey norm assumptions.

Coulomb Gauges and Regularity for Stationary Weak Yang$-$Mills Connections in Supercritical Dimension

TL;DR

This work addresses regularity for stationary weak Yang–Mills connections in the supercritical dimension within the weak-connection framework. It develops a Morrey-norm based Coulomb-gauge extraction and a two-stage approximation scheme that smoothe weak connections into smooth ones while preserving controlled Morrey norms of the curvature. The authors prove an epsilon-regularity theorem and deduce partial regularity, establishing that the singular set has vanishing -dimensional Hausdorff measure, thereby extending prior results for smooth, Sobolev, and admissible connections to the broader weak-connection setting. The methods rely on extension and gluing techniques on cubes, admissible covers, and careful trace control, significantly advancing the understanding of Yang–Mills fields in dimensions above the critical threshold and enabling robust bubbling and energy-quantization analyses.

Abstract

We prove that stationary YangMills fields in dimensions 5 belonging to the variational class of weak connections are smooth away from a closed singular set of vanishing 1-dimensional Hausdorff measure. Our proof is based on an -regularity theorem, which generalizes to this class of weak connections the existing previous -regularity results by G. Tian for smooth connections, by Y. Meyer and the second author for Sobolev and approximable connections, and by T. Tao and G. Tian for admissible connections (which are weak limits of smooth YangMills fields). On the path towards establishing -regularity, a pivotal step is the construction of controlled Coulomb gauges for general weak connections under small Morrey norm assumptions.

Paper Structure

This paper contains 14 sections, 33 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. The building blocks for the approximation theorems
  3. Extension of weak connections
  4. Extension under Z-smallness of the connection and its curvature
  5. Extension under Z-smallness of the curvature only
  6. Construction of optimally regular gauges on the boundary of cubes
  7. The notion of admissible covers and good and bad cubes
  8. The notion of admissible cubic Z-covers
  9. The notion of good and bad cubes
  10. The first smoothification
  11. The second smoothification: strong Z-approximation by smooth connections
  12. Partial regularity for stationary weak Yang--Mills fields
  13. Adapted Coulomb gauge extraction statements in critical dimension
  14. Z-valued map extensions of traces in Z

Key Result

Theorem I.1

petrache-riviere-global-gauges Let $\nabla\in{\mathbb A}_{\operatorname{SU}(2)}(N^n)$ be a Sobolev connection of some principal $\operatorname{SU}(2)$-bundle over a closed oriented Riemannian manifold $N^n$ of dimension $n\leqslant 4$. Then there exists $A\in L^{4,\infty}(\wedge^1N^n\otimes {\mathfr where $L^{4,\infty}$ is the weak Marcinkiewicz space with $|\cdot|$ being the measure induced by t

Theorems & Definitions (71)

  • Theorem I.1
  • Definition I.1: Weak Yang--Mills connections
  • Definition I.2: Stationary weak Yang--Mills connections
  • Theorem I.2
  • Theorem I.3: $\varepsilon$-regularity
  • Theorem I.4
  • Proposition II.1: Harmonic extension under smallness condition on $F_A$ and $A$
  • proof : Proof of Proposition \ref{['Proposition: harmonic extension on the good cubes']}
  • Corollary II.1
  • proof : Proof of Corollary \ref{['Corollary: extension in the interior of good cubes']}
  • ...and 61 more