Sheeting property and rigidity for stable solutions to the Allen--Cahn equation

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 $\epsilon \mathrm{\mid }D{u}_{\epsilon }{\mathrm{\mid }}^{2}dX$ to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in ${\mathbb{R}}^N$, $N\le 7$, with $O(R^{N-1})$ energy growth are one-dimensional. When $W^{\mathrm{\prime }\mathrm{\prime }}(-1)=W^{\mathrm{\prime }\mathrm{\prime }}(1)$, every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for $4\le N\le 10$. 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.