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

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On a compact manifold Mⁿ (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.