We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $\geq 2$. We investigate the limiting behavior of these critical points as $\varepsilon \to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $\varepsilon \to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.
"Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds." J. Differential Geom. 118 (2) 335 - 371, June 2021. https://doi.org/10.4310/jdg/1622743143