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

TL;DR

This paper proves that for certain critical points of the Ginzburg-Landau energy on manifolds with non-zero first Betti number, energy concentrates on a stationary varifold as the parameter tends to zero, revealing geometric structure.

Contribution

It introduces a new estimate for Ginzburg-Landau energies on manifolds with non-zero first Betti number, linking energy concentration to stationary varifolds in the limit.

Findings

Energy concentrates on a stationary, rectifiable (n-2)-varifold as epsilon approaches zero.

Decomposition of harmonic component of the Jacobian is key to the estimate.

Critical points from min-max construction exhibit energy concentration phenomena.

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 $\epsilon\to 0$.