发表机构
Beijing Institute of Technology（北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明闭黎曼流形上各向异性Allen-Cahn方程中，严格F-稳定的F-极小超曲面作为平滑极限界面时重数为一，采用障碍与滑动方法并借助能量估计。
AI 中文摘要
本文研究闭黎曼流形上各向异性Allen-Cahn方程的平滑极限界面的重数问题。我们证明，当孤立平滑极限界面是闭的、嵌入的、双侧的严格F-稳定F-极小超曲面时，其重数为一。证明采用各向异性障碍和滑动论证，将节点集限制在宽度为O(ε)的邻域内。随后我们将非恒定缩放极限分类为一维解，并在极限超曲面附近获得能量上界。指数衰减控制剩余能量，从而得出重数为一的结论。
英文摘要
In this paper, we study the multiplicity of smooth limit interfaces for the anisotropic Allen-Cahn equation on closed Riemannian manifolds. We prove that an isolated smooth limit interface has multiplicity one whenever it is a closed, embedded, two-sided, strictly \(F\)-stable \(F\)-minimal hypersurface. The proof uses anisotropic barriers and a sliding argument to confine the nodal sets to a neighborhood of width \(O(\varepsilon)\). We then classify the nonconstant rescaled limits as one-dimensional solutions and obtain an energy upper bound near the limiting hypersurface. Exponential decay controls the remaining energy, yielding the multiplicity-one conclusion.
