Asymptotics of finite energy monopoles on AC $3$-manifolds

TL;DR

This paper analyzes the asymptotic behavior of finite energy SU(2) monopoles on asymptotically conical 3-manifolds, extending classical results from Euclidean space and establishing decay rates and convergence properties of critical points.

Contribution

It generalizes classical monopole results to AC 3-manifolds, proving integrality of monopole charge, energy formulas, and decay properties of critical points.

Findings

Monopole charge is integral.

Critical points' Higgs field derivatives decay quadratically.

Curvature decays exponentially under certain conditions.

Abstract

We study the asymptotic behavior of finite energy $\rm{SU}(2)$ monopoles, and general critical points of the $\rm{SU}(2)$ Yang--Mills--Higgs energy, on asymptotically conical $3$-manifolds with only one end. Our main results generalize classical results due to Groisser and Taubes in the particular case of the flat $3$-dimensional Euclidean space $\mathbb{R}^3$. Indeed, we prove the integrality of the monopole number, or charge, of finite energy configurations, and derive the classical energy formula establishing monopoles as absolute minima. Moreover, we prove that the covariant derivative of the Higgs field of a critical point of the energy decays quadratically along the end, and that its transverse component with respect to the Higgs field, as well as the corresponding component of the curvature of the underlying connection, actually decay exponentially. Additionally, under the assumption of positive Gaussian curvature on the asymptotic link, we prove that the curvature of any critical point connection decays quadratically. Furthermore, we deduce that any irreducible critical point converges uniformly along the conical end to a limiting configuration at infinity consisting of a reducible Yang--Mills connection and a parallel Higgs field.