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Convergence of the self-dual abelian Higgs gradient flow

Jason Zhao

Abstract

Given an initial data configuration $(A^{\mathrm{in}}, φ^{\mathrm{in}})$ on $\mathbb R^2$ such that the self-dual abelian Higgs energy is near the minimum energy within its topological class, we prove that its evolution under the self-dual abelian Higgs gradient flow in temporal gauge converges exponentially as $t \to \infty$ with respect to the $(H^1 \times L^2)$-metric to a minimiser of the energy. Furthermore, we show that the convergence of the scalar field $φ$ may be upgraded to the $H^1$-metric provided the additional assumption on the potential that $A^{\mathrm{in}} \in L^p (\mathbb R^2)$ for $2 < p < \infty$. As a corollary, we obtain a quantitative stability for the self-dual abelian Higgs energy which improves upon the previous result of Halavati (arXiv:2310.04866) and partially resolves the open problem posed in his article.

Convergence of the self-dual abelian Higgs gradient flow

Abstract

Given an initial data configuration on such that the self-dual abelian Higgs energy is near the minimum energy within its topological class, we prove that its evolution under the self-dual abelian Higgs gradient flow in temporal gauge converges exponentially as with respect to the -metric to a minimiser of the energy. Furthermore, we show that the convergence of the scalar field may be upgraded to the -metric provided the additional assumption on the potential that for . As a corollary, we obtain a quantitative stability for the self-dual abelian Higgs energy which improves upon the previous result of Halavati (arXiv:2310.04866) and partially resolves the open problem posed in his article.

Paper Structure

This paper contains 17 sections, 33 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Outline of the article
  4. Preliminaries
  5. Notation
  6. Covariant identities and inequalities
  7. Magnetic Sobolev estimates
  8. Topological degree
  9. Self-dual structure
  10. Covariant smoothing estimates
  11. Convergence of the flow
  12. Decay of Bogomoln'yi tension field
  13. $(H^1\times L^2)$-convergence of $(A, \phi)$
  14. $\dot H^1$-convergence of scalar field
  15. Well-posedness theory
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $N \in \mathbb Z$ be an integer, and suppose $\varepsilon_* \ll_{|N|} 1$ and $\gamma \ll_{|N|} 1$ are sufficiently small, and $C \gg_{|N|} 1$ is sufficiently large. Consider a finite-energy configuration $(A^{\mathrm{in}}, \phi^{\mathrm{in}})$ on $\mathbb R^2$ with topological degree $N$ and sel and let $(A, \phi)$ be the solution to the self-dual abelian Higgs gradient flow (eq:AHG) on $[0, \

Theorems & Definitions (74)

  • Theorem 1.1: Convergence of self-dual flow to $\mathcal{M}^N$
  • Remark
  • Remark
  • Remark
  • Corollary 1.2: Stability of the self-dual energy
  • Remark
  • Lemma 2.1: Curvature identities
  • proof
  • Lemma 2.2: Covariant Laplacian commutator identity
  • proof
  • ...and 64 more