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A note on weak compactness of $Ω$-Yang-Mills connections

Chang-Yu Guo, Chang-Lin Xiang

TL;DR

This work extends the weak compactness result for Yang-Mills connections to the broader class of $Ω$-Yang-Mills connections on compact Riemannian manifolds. Under an $L^2$-bound on the curvature $F_A$ and weak $L^4_{\mathrm{loc}}$ convergence of the connection 1-forms, the limit of a sequence of weak $Ω$-YM connections remains a weak $Ω$-YM connection, establishing weak continuity in this setting. The authors combine distributional convergence of the curvature with a compensated compactness framework for the nonlinear term $\ast(F_A\wedge\Omega)$ and employ a Chen–Giron-type weak continuity lemma to pass to the limit in the Euler–Lagrange expressions. This result provides a foundation for extending singularity stratification and energy-identity analyses, such as those developed by Naber–Valtorta, to $Ω$-YM fields and informs future regularity and compactness studies in $Ω$-YM theory.

Abstract

In this note, applying a compensation compactness argument developped by Chen and Giron (arXiv.2108.13529) on Yang-Mills fields, we extends their weak continuity result to the more general class of $Ω$-Yang-Mills connections on principle bundles over compact Riemannian manifold.

A note on weak compactness of $Ω$-Yang-Mills connections

TL;DR

This work extends the weak compactness result for Yang-Mills connections to the broader class of -Yang-Mills connections on compact Riemannian manifolds. Under an -bound on the curvature and weak convergence of the connection 1-forms, the limit of a sequence of weak -YM connections remains a weak -YM connection, establishing weak continuity in this setting. The authors combine distributional convergence of the curvature with a compensated compactness framework for the nonlinear term and employ a Chen–Giron-type weak continuity lemma to pass to the limit in the Euler–Lagrange expressions. This result provides a foundation for extending singularity stratification and energy-identity analyses, such as those developed by Naber–Valtorta, to -YM fields and informs future regularity and compactness studies in -YM theory.

Abstract

In this note, applying a compensation compactness argument developped by Chen and Giron (arXiv.2108.13529) on Yang-Mills fields, we extends their weak continuity result to the more general class of -Yang-Mills connections on principle bundles over compact Riemannian manifold.

Paper Structure

This paper contains 2 sections, 3 theorems, 40 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of main result

Key Result

Theorem 1.2

Let $\tilde{\nabla}$ be a fixed reference connection on the principle bundle $P\to M$ and $\nabla_{A_{i}}=\tilde{\nabla}+A_{i}\in{\mathcal{A}}_{\text{\rm loc}}^{0,4}(P)$ a sequence of weak $\Omega$-Yang-Mills connections. Assume that and Then $\tilde{\nabla}+A$ is a weak $\Omega$-YM connection as well.

Theorems & Definitions (6)

  • Definition 1.1
  • Theorem 1.2: Weak compactness
  • Lemma 2.1
  • proof
  • Lemma 2.2: Weak continuity, Chen-Giron-21
  • proof