Multiplicity one and strictly stable Allen-Cahn minimal hypersurfaces

Abstract

We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work in all dimensions and without variational assumptions on the Allen-Cahn solutions.

arXiv comment29 pages. We have improved our main result to a local version. Comments are welcome

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Cited by in corpus (5)

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• Rigidity of low index solutions on $S^3$ via a Frankel theorem for the Allen-Cahn equation
• Ground states of semilinear elliptic equations
• Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn