Quantitative stability of Yang-Mills-Higgs instantons in two dimensions

TL;DR

This paper establishes the second order stability of nearly minimizing Yang-Mills-Higgs vortex pairs in two dimensions, using new inequalities, compactness, and perturbation methods, with extensions to arbitrary surfaces.

Contribution

It introduces a novel approach combining weighted inequalities and a selection principle to prove stability of vortex pairs, extending results to general line bundles over surfaces.

Findings

Nearly minimizing pairs are second order close to true minimizers.

Stability results are generalized to arbitrary compact surfaces.

New weighted inequalities facilitate the stability analysis.

Abstract

We prove that if an N-vortex pair nearly minimizes the Yang-Mills-Higgs energy, then it is second order close to a minimizer. First we use new weighted inequalities in two dimensions and compactness arguments to show stability for sections with some regularity. Second we define a selection principle using a penalized functional and by elliptic regularity and smooth perturbation of complex polynomials, we generalize the stability to all nearly minimizing pairs. With the same method, we also prove the analogous second order stability for nearly minimizing pairs on nontrivial line bundles over arbitrary compact smooth surfaces.