Convergence of Yang–Mills–Higgs fields

Abstract

In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces, where the fiber is a compact symplectic manifold and the conformal structure of the underlying surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth YMH field modulo finitely many harmonic spheres, while near the nodes where the conformal structure degenerates, the YMH fields converges to a pair consisting of a flat connection and a twisted geodesic (with potential). In particular, we generalize the recent compactness results on both harmonic maps from surfaces and twisted holomorphic curves to general YMH fields.

Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

“…the Poincaré inequality with connection on S 1 . The inequality is already proved and played an important role in the blow-up analysis of a sequence of YMH fields in [10], here we take a step further by exploring the best constant of the Poincaré inequality.…”

“…This motivates us to choose the so-called balanced temporal gauge, whose existence is guaranteed by the following lemma, cf. Lemma 3.2 in [10].…”

“…In this paper, we study two dimensional YMH fields with isolated singularities, where Σ is a Riemann surface. This type of singular YMH fields naturally emerges as the limit of a sequence of two dimensional YMH fields with finite energy, if we allow the conformal structure of the underlying surface to vary, see [10].…”

“…This type of bubbles turns out to be twisted harmonic maps instead of usual harmonic maps, again due to possibly non-trivial holonomy along the shrinking geodesic. See [10] for more details. Theorem 1.2.…”

“…u tt + ∂ 2 θ,α u = Γ(u)(∇ A u, ∇ A u) + f.Then a standard argument yields the following ε-regularity for u (cf. Lemma 4.1 in[10]). …”