Minimal submanifolds from the abelian Higgs model

Abstract

Given a Hermitian line bundle L→ M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as ϵ→ 0 , of couples (uϵ, ∇ ϵ) critical for the rescalings Eϵ(u,∇)=∫M(|∇u|2+ϵ2|F∇|2+14ϵ2(1-|u|2)2)of the self-dual Yang–Mills–Higgs energy, where u is a section of L and ∇ is a Hermitian connection on L with curvature F∇. Under the natural assumption lim sup ϵ→Eϵ(uϵ, ∇ ϵ) < ∞, we show that the energy measures converge subsequentially to (the weight measure μ of) a stationary integral (n- 2) -varifold. Also, we show that the (n- 2) -currents dual to the curvature forms converge subsequentially to 2 πΓ , for an integral (n- 2) -cycle Γ with | Γ | ≤ μ. Finally, we provide a variational construction of nontrivial critical points (uϵ, ∇ ϵ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral (n- 2) -varifolds in an arbitrary closed Riemannian manifold.