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

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

2010-06-17

arxiv.org/abs/1006.3356v1

Mathematics

Analysis of PDEs

accepted in GAFA

Scientific paper

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

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