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Isolated singularities of 3-dimensional Yang-Mills-Higgs fields

Bo Chen, Chong Song

Abstract

In this paper, we derive decay estimates near isolated singularities of 3-dimensional (3d) Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a connected compact Lie group. As an application, we obtain removable singularity theorems for 3d Yang-Mills-Higgs fields under different types of energy conditions, which generalizes classical removable singularity theorems for 3d Yang-Mills fields~\cite{S84,SS84} and 3d harmonic maps~\cite{L85}.

Isolated singularities of 3-dimensional Yang-Mills-Higgs fields

Abstract

In this paper, we derive decay estimates near isolated singularities of 3-dimensional (3d) Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a connected compact Lie group. As an application, we obtain removable singularity theorems for 3d Yang-Mills-Higgs fields under different types of energy conditions, which generalizes classical removable singularity theorems for 3d Yang-Mills fields~\cite{S84,SS84} and 3d harmonic maps~\cite{L85}.

Paper Structure

This paper contains 20 sections, 25 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. The Yang-Mills-Higgs functional and backgrounds
  3. YMH fields with isolated singularities
  4. Preliminaries
  5. Equations for YMH fields
  6. Bochner formula for YMH fields
  7. $\varepsilon$-regularity theorems
  8. Kato inequalities
  9. Kato inequalities for 1-forms
  10. Kato inequalities for 2-forms
  11. Improved Kato inequalities for YMH fields
  12. Differential inequalities of 3d YMH fields
  13. Decay estimates of 3d YMH fields: energy bound I
  14. Improved regularity of $F_A$
  15. Proof of Theorem \ref{['decay-YMH']}: energy bound I
  16. ...and 5 more sections

Key Result

Theorem 1.1

There exists an $\varepsilon_0>0$ such that if $(A,u)$ is a YMH field on $B^*_{R_0}\subset \mathbb{R}^3$ with $R_0\leq 1$, which satisfies one of the following energy bounds then there is a $0<r_0\leq R_0/2$ such that $(A,u)$ satisfies and for any $r=|x|\leq r_0$, where $\alpha=\sqrt{\frac{1}{4}-C(\varepsilon^2_0+r^{2/3}_0)}$. Here the constant $C$ in main-theorem-curvature and main-theorem-sec

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • ...and 32 more