Multiplicity One for Strictly Stable Anisotropic Allen-Cahn Minimal Hypersurfaces

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 $O(\epsilon )$. 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.