Convergence of the self-dual abelian Higgs gradient flow

Abstract

Given an initial data configuration $(A^{\mathrm{in}},\ddot{I}{•}^{\mathrm{in}})$ 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 $\ddot{I}•$ 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.

27 page, no figures; fixed typos