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.
