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 \(E_{\epsilon}(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4\epsilon^2}(1-|u|^2)^2\) 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 \(u_{\epsilon}\) of \(E_{\epsilon}\) 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 \(\epsilon\to 0\).