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 $${O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}$$ .