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

On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $\epsilon$ tends to zero of solutions $u_{\epsilon}: M \to \mathbb{C}$ to the equation $\Delta u_{\epsilon} + \epsilon^{-2}(1 - |u_{\epsilon}|^{2})u_{\epsilon} = 0$ with the boundary condition $\partial_{\nu}u_{\epsilon} = 0$ on $\partial M$. Assuming an energy upper bound on the solutions and a convexity condition on $\partial M$, we show that along a subsequence, the energy of $\{u_{\epsilon}\}$ breaks into two parts: one captured by a harmonic $1$-form $\psi$ on $M$, and the other concentrating on the support of a rectifiable $(n-2)$-varifold $V$ which is stationary with respect to deformations preserving $\partial M$. Examples are given which shows that $V$ could vanish altogether, or be non-zero but supported only on $\partial M$.