We use a min-max procedure on the Allen–Cahn energy functional to construct geodesics on closed, $2$‑dimensional Riemannian manifolds, as motivated by the work of Guaraco [Gua18]. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen–Cahn curvature estimates due to Wang–Wei [WW19], we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding “entire” singularity models constructed by del Pino–Kowalczyk–Pacard [dPKP13] with Morse index $1$. The argument is completed by a Morse index estimate on these singularity models.
"Allen–Cahn min-max on surfaces." J. Differential Geom. 117 (1) 93 - 135, January 2021. https://doi.org/10.4310/jdg/1609902018