Asymptotics for the Ginzburg-Landau equation on manifolds with boundary under homogeneous Neumann condition

Abstract

On a compact manifold M‾n (n≥3) with boundary, we study the asymptotic behavior as ϵ tends to zero of solutions uϵ:M→C to the equation Δuϵ+ϵ−2(1−|uϵ|2)uϵ=0 with the boundary condition ∂νuϵ=0 on ∂M. Assuming an energy upper bound on the solutions and a convexity condition on ∂M, we show that along a subsequence, the energy of {uϵ} breaks into two parts: one captured by a harmonic 1-form ψ on M, and the other concentrating on the support of a rectifiable (n−2)-varifold V which is stationary with respect to deformations preserving ∂M. Examples are given which shows that V could vanish altogether, or be non-zero but supported only on ∂M.