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

TL;DR

This paper proves the existence of multi-layer solutions to the Allen-Cahn equation near minimal submanifolds in positively Ricci curved Riemannian manifolds, with layers concentrating along the submanifold as the perturbation parameter tends to zero.

Contribution

It establishes the existence of solutions with multiple transition layers near minimal submanifolds under positive Ricci curvature conditions, extending understanding of phase transition interfaces.

Findings

Solutions with multiple layers near minimal submanifolds are constructed.

Transition layers are separated by a distance proportional to psilon |ln psilon|.

The solutions concentrate along the minimal submanifold as psilon .

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 $O(\epsilon |\ln \epsilon|)$.