A Natural Min-Max Construction for Ginzburg-Landau Functionals

Abstract

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these solutions concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold $V$.

changes from v2: added theorem 1.2 and section 5 about energy concentration varifold when $H^1_{dR}(M)=0$; added references; fixed typos