Energy Concentration for Min-Max Solutions of the Ginzburg-Landau Equations on Manifolds with $b_1(M)\neq 0$

Abstract

We establish a new estimate for the Ginzburg-Landau energies $E_ε(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4ε^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^1du^2-u^2du^1$ into an integral and fractional part. We employ this estimate to show that, for critical points $u_ε$ of $E_ε$ arising from the two-parameter min-max construction considered by the author in previous work, a nontrivial portion of the energy must concentrate on a stationary, rectifiable $(n-2)$-varifold as $ε\to 0$.