A Natural Min-Max Construction for Ginzburg-Landau Functionals

Abstract

We use min-max techniques to produce nontrivial solutions $u_{\epsilon }:M\to {\mathbb{R}}^2$ of the Ginzburg-Landau equation $\Delta u_{\epsilon }+\frac{1}{{\epsilon }^2}(1-|u_{\epsilon }{|}^2)u_{\epsilon }=0$ on a given compact Riemannian manifold, whose energy grows like $|\log \epsilon |$ as $\epsilon \to 0$. When the degree one cohomology $H_{dR}^1(M)=0$, 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 $H_{dR}^1(M)=0$; added references; fixed typos