Stability and energy identity for Yang–Mills–Higgs pairs

Abstract

In this paper, we study the properties of the critical points of Yang–Mills–Higgs functional, which are called Yang–Mills–Higgs pairs. We first consider the properties of weakly stable Yang–Mills–Higgs pairs on a vector bundle over Sn (n ≥ 4). When n ≥ 4, we prove that the norm of its Higgs field is 1 and the connection is actually Yang–Mills. More precisely, its curvature vanishes when n ≥ 5. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang–Mills–Higgs pairs over a four-dimensional compact manifold with uniformly bounded energy. We show there is a subsequence that converges smoothly to a Yang–Mills–Higgs pair up to gauge modulo finitely many four-dimensional spheres with Yang–Mills connections.