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PreprintarXiv licencearXiv

Multiplicity One for Strictly Stable Anisotropic Allen-Cahn Minimal Hypersurfaces

  • Lu Chen
  • Jiali Lan
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Abstract

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(ε). 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.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026multiplicity,
  title = {{Multiplicity One for Strictly Stable Anisotropic Allen-Cahn Minimal Hypersurfaces}},
  author = {Lu Chen and Jiali Lan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06981},
  eprint = {2610.06981},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06981},
}
RIS
TY  - UNPB
TI  - Multiplicity One for Strictly Stable Anisotropic Allen-Cahn Minimal Hypersurfaces
AU  - Lu Chen
AU  - Jiali Lan
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.06981
UR  - https://arxiv.org/abs/2610.06981
AB  - 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(ε). 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.
ER  -