A Natural Min-Max Construction for Ginzburg-Landau Functionals

We use min-max techniques to produce nontrivial solutions $u_{\epsilon}:M\to $\mathbb{R}$^2$oftheGinzburg-Landauequation$\Delta u_{\epsilon}+\frac{1}{\epsilon^2}(1-|u_{\epsilon}|^2)u_{\epsilon}=0$ on a given compact Riemannian manifold, whose energy grows like $\mathrm{\mid }\log ϵ\mathrm{\mid }$ as $ϵ\to 0$. 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 $V$.