发表机构
The Chinese University of Hong Kong; University of Warwick; University of Toronto（香港中文大学; 华威大学; 多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Allen-Cahn方程研究带边黎曼流形中自由边界极小子流形的极小极大理论，证明稳定解的极限界面为满足广义稳定性不等式的带边界曲率varifold，并证实边界集中例子具有无界Morse指标。
AI 中文摘要
这是我们系列工作的第二篇论文，旨在通过Allen-Cahn方程发展带边黎曼流形中自由边界极小子流形的极小极大理论。基于我们之前关于一般临界点的工作，我们研究了由带Neumann边界条件的稳定（或有限Morse指标）Allen-Cahn方程解产生的极限界面积分varifold的边界行为。更精确地说，我们证明了极限界面是Mantegazza意义下的带边界曲率varifold，满足广义稳定性不等式。此外，我们排除了边界处的经典切锥，并证明了极限界面varifold的第一变分关于其权重测度是奇异的。特别地，我们的结果证实了一个民间猜想：Malchiodi、Ni和Wei的边界集中例子具有无界Morse指标。正如我们即将发表的工作中将显而易见的，本文获得的结果是Allen-Cahn自由边界极小极大理论中达到最优边界正则性的关键要素。
英文摘要
This is the second paper in our series of works to develop the min-max theory for free boundary minimal hypersurfaces in Riemannian manifolds with boundary via the Allen-Cahn equation. Building on our previous work for general critical points, we study the boundary behaviour of limit-interface integral varifolds arising from stable (or finite Morse index) solutions to the Allen-Cahn equation with Neumann boundary condition. More precisely, we prove that the limit-interface is a curvature varifold with boundary in the sense of Mantegazza satisfying a generalized stability inequality. Furthermore, we rule out classical tangent cones at the boundary and show that the first variation of the limit-interface varifold is singular with respect to its weight measure. In particular, our result confirms a folklore conjecture that the boundary concentration examples of Malchiodi, Ni and Wei have unbounded Morse index. As will be evident in our forthcoming work, the results obtained in this paper are crucial ingredients towards the optimal boundary regularity in the Allen-Cahn free boundary min-max theory.
