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. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, 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 with Morse index 1. The argument is completed by a conjecturally sharp Morse index estimate on these singularity models.

中文翻译：

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表面上的 Allen-Cahn min-max

我们在 Allen-Cahn 能量泛函上使用 min-max 程序在封闭的二维黎曼流形上构建测地线，这是由 Guaraco 的工作所激发的。借用几何分析中经典的膨胀和曲率估计，以及由 Wang-Wei 提出的新的 Allen-Cahn 曲率估计，我们设法研究了漫反射级潜在奇异点的精细结构，并表明问题简化为理解由 del Pino-Kowalczyk-Pacard 构建的具有莫尔斯指数 1 的“整个”奇点模型。通过对这些奇点模型的推测性尖锐的莫尔斯指数估计来完成论证。