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.

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

References in corpus (3)

• New applications of Min-max Theory
• Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in ${\mathbb{R}}^3$
• On De Giorgi Conjecture in Dimension $N\ge 9$

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• Variational aspects of phase transitions with prescribed mean curvature
• 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