Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable solutions of $U(1)$ Yang-Mills-Higgs model in $\mathbb{R}^4$

Yong Liu, Juncheng Wei, Zikai Ye

Abstract

We give a positive answer to the conjecture of Liu-Ma-Wei-Wu in \cite{LMWW} that the family of entire solutions to the $U(1)$-Yang-Mills-Higgs equation constructed by the gluing method in that paper are stable. This is the first family of examples of nontrivial stable critical points to the $U(1)$-Yang-Mills-Higgs model in higher dimensional Euclidean space. Intuitively, the stability of these solutions corresponds to the fact that holomorphic curves are area-minimizing. We also show that these entire solutions are non-degenerate. Our proof is based on detailed analysis of the linearized operators around this family and the spectrum estimates of the Jacobi operator by Arezzo-Pacard \cite{ArePac}.

Stable solutions of $U(1)$ Yang-Mills-Higgs model in $\mathbb{R}^4$

Abstract

We give a positive answer to the conjecture of Liu-Ma-Wei-Wu in \cite{LMWW} that the family of entire solutions to the -Yang-Mills-Higgs equation constructed by the gluing method in that paper are stable. This is the first family of examples of nontrivial stable critical points to the -Yang-Mills-Higgs model in higher dimensional Euclidean space. Intuitively, the stability of these solutions corresponds to the fact that holomorphic curves are area-minimizing. We also show that these entire solutions are non-degenerate. Our proof is based on detailed analysis of the linearized operators around this family and the spectrum estimates of the Jacobi operator by Arezzo-Pacard \cite{ArePac}.

Paper Structure

This paper contains 8 sections, 15 theorems, 204 equations.

Table of Contents

  1. Introduction and statement of main result
  2. Preliminaries
  3. The minimal submanifold $\Gamma$
  4. Vortex solutions
  5. Eigenvalue problems and apriori estimates
  6. Improvement of approximation
  7. Analysis of the linearized operator
  8. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

The solutions $U_{\epsilon}$ are stable and non-degenerate.

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.4.1
  • Definition 2
  • Lemma 2.5
  • Lemma 2.6
  • ...and 14 more