Interface foliation near a minimal isoparametric hypersurface for the Allen-Cahn equation

Let $(M,g)$ be a closed Riemannian manifold of positive Ricci curvature. Let $f$ be a proper isoparametric function on $M$, and $\mathrm{\Gamma }$ the unique level set of $f$ which is a minimal hypersurface. For any positive integer $k$, and $\lambda >0$ large enough, we construct a solution of the Allen-Cahn equation $\mathrm{\Delta }u+\lambda (u-u^3)=0$ which is constant along the level sets of $f$ and has exactly $k$ nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface $\mathrm{\Gamma }$ as $\lambda \to \mathrm{\infty }$ and the energy is uniformly bounded for all such solutions.