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

Abstract

Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ \epsilon^2\Delta_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where $\epsilon$ is a small parameter. Let $\KK\subset \MM$ be an $(N-1)$-dimensional smooth minimal submanifold that separates $\MM$ into two disjoint components. Assume that $\KK$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that $|A_{\KK}|^2+\mbox{Ric}_{\tilde g}(\nu_{\KK}, \nu_{\KK})$ is positive along $\KK$. Then for each integer $m\geq 2$, we establish the existence of a sequence $\epsilon = \epsilon_j\to 0$, and solutions $u_{\epsilon}$ with $m$-transition layers near $\KK$, with mutual distance $O(\epsilon |\ln \epsilon|)$.