Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let (M,g̃) be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation where ε is a small parameter. Let K ⊂ M be an (N - 1)-dimensional smooth minimal submanifold that separates M into two disjoint components. Assume that K is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that is positive along K. Then for each integer m ≥ 2, we establish the existence of a sequence ε = ε_j → 0, and solutions u_ε with m-transition layers near K, with mutual distance O(ε{pipe}lnε{pipe}).