We use min-max techniques to produce nontrivial solutions $u_{epsilon}:Mto mathbb{R}^2$ 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 $|logepsilon|$ as $epsilonto 0$. 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 $V$.
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