Allen–Cahn min-max on surfaces

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