Boundary behaviour of stable Allen-Cahn limit-interfaces in Riemannian manifolds

Abstract

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.

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BibTeX RIS

BibTeX

@misc{li2026boundary,
  title = {{Boundary behaviour of stable Allen-Cahn limit-interfaces in Riemannian manifolds}},
  author = {Martin Man-chun Li and Davide Parise and Lorenzo Sarnataro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04833},
  eprint = {2610.04833},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04833},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Boundary behaviour of stable Allen-Cahn limit-interfaces in Riemannian manifolds
AU  - Martin Man-chun Li
AU  - Davide Parise
AU  - Lorenzo Sarnataro
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.04833
UR  - https://arxiv.org/abs/2610.04833
AB  - 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.
ER  -

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