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

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 $b_1(M)\ne 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^{1}d{u}^2-u^{2}d{u}^1$ 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 $\epsilon \to 0$.

References in corpus (1)

• A Natural Min-Max Construction for Ginzburg-Landau Functionals