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

Abstract

Let ${(\mathcal {M},\tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation

$$\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},$$

where ${\varepsilon}$ is a small parameter. Let ${{\mathcal {K} \subset \mathcal {M}}}$ be an (N - 1)-dimensional smooth minimal submanifold that separates ${\mathcal{M}}$ into two disjoint components. Assume that ${\mathcal{K}}$ is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that ${{|A_\mathcal {K} |^2 + {\rm {Ric}}_{\tilde g} (v _{\mathcal K}, v_{\mathcal K})}}$ is positive along ${\mathcal{K}}$. Then for each integer m ≥ 2, we establish the existence of a sequence ${\varepsilon = \varepsilon{_j} \rightarrow 0}$, and solutions ${u_\varepsilon}$ with m-transition layers near ${\mathcal{K}}$, with mutual distance ${O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}$.