PreprintarXiv licencearXiv
Interface foliation near a minimal isoparametric hypersurface for the Allen-Cahn equation
Abstract
Let (M,g) be a closed Riemannian manifold of positive Ricci curvature. Let f be a proper isoparametric function on M, and Γ the unique level set of f which is a minimal hypersurface. For any positive integer k, and λ>0 large enough, we construct a solution of the Allen-Cahn equation $ Δu + λ(u-u^3 ) =0$ which is constant along the level sets of f and has exactly k nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface Γ as λ→ ∞ and the energy is uniformly bounded for all such solutions.
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BibTeX
@misc{henry2026interface,
title = {{Interface foliation near a minimal isoparametric hypersurface for the Allen-Cahn equation}},
author = {Guillermo Henry and Jimmy Petean},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05715},
eprint = {2609.05715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05715},
}
RIS
TY - UNPB
TI - Interface foliation near a minimal isoparametric hypersurface for the Allen-Cahn equation
AU - Guillermo Henry
AU - Jimmy Petean
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05715
UR - https://arxiv.org/abs/2609.05715
AB - Let (M,g) be a closed Riemannian manifold of positive Ricci curvature. Let f be a proper isoparametric function on M, and Γ the unique level set of f which is a minimal hypersurface. For any positive integer k, and λ>0 large enough, we construct a solution of the Allen-Cahn equation $ Δu + λ(u-u^3 ) =0$ which is constant along the level sets of f and has exactly k nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface Γ as λ→ ∞ and the energy is uniformly bounded for all such solutions.
ER -