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

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