Allen–Cahn min-max on surfaces
Allen–Cahn min-max on surfaces
复制标题
DOI：
10.4310/jdg/1609902018
复制
发表时间：
2017-06
影响因子：
2.5
通讯作者：
Christos Mantoulidis
中科院分区：
文献类型：
--
作者：
Christos Mantoulidis
受Guaraco的启发，我们在Allen-Cahn能量泛函上使用最小-最大过程在封闭的二维黎曼流形上构造测地线。借用几何分析中的经典爆破和曲率估计，以及王伟提出的新颖的Allen-Cahn曲率估计，我们成功地在扩散水平上研究了潜在奇点的精细结构，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。通过对这些奇异模型的莫尔斯指数估计完成了论证。
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 Morse index estimate on these singularity models.