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Sheeting property and rigidity for stable solutions to the Allen--Cahn equation
Abstract
We prove a sheeting theorem for stable solutions to the Allen--Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures ε|Duε|2 dX to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in ℝN, N≤7, with O(RN−1) energy growth are one-dimensional. When W''(−1)=W''(1), every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for 4≤ N≤10. For even potentials, we also obtain multiplicity one for limits with bounded energy and Morse index on closed manifolds of dimensions $3$ through $7$, under bumpiness or positive Ricci curvature.
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BibTeX
@misc{liu2026sheeting,
title = {{Sheeting property and rigidity for stable solutions to the Allen--Cahn equation}},
author = {Yong Liu and Tianci Luo and Kelei Wang and Juncheng Wei and Yong Wei and Ke Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08499},
eprint = {2610.08499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08499},
}
RIS
TY - UNPB
TI - Sheeting property and rigidity for stable solutions to the Allen--Cahn equation
AU - Yong Liu
AU - Tianci Luo
AU - Kelei Wang
AU - Juncheng Wei
AU - Yong Wei
AU - Ke Wu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08499
UR - https://arxiv.org/abs/2610.08499
AB - We prove a sheeting theorem for stable solutions to the Allen--Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures ε|Du_ε|^2 dX to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in ℝ^N, N≤7, with O(R^N−1) energy growth are one-dimensional. When W''(−1)=W''(1), every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for 4≤ N≤10. For even potentials, we also obtain multiplicity one for limits with bounded energy and Morse index on closed manifolds of dimensions $3$ through $7$, under bumpiness or positive Ricci curvature.
ER -