Title：Interface foliation near minimal submanifolds in Riemannian manifolds with Positive Ricci curvature

Speaker：杨 军  教授  （华中师范大学）

Time & Room：

      9月20日, 09:00-12:00, 1318 教室;

      9月21日, 14:00-17:00, 1218 教室; 

      9月22日, 09:00-12:00, 1218 教室.

Abstract：We will talk about the construction of clustering interfaces for Allen-Cahn equation on compact Riemannian manifold by the infinitely dimensional reduction method(from "M. del Pino, M. Kowalczyk, J. Wei, Comm. Pure Appl. Math. 2007").  The interaction of neighbouring interfaces will be derived in a form of Toda-Jacobi System involving a resonance phenomena. The result appeared in "M. del Pino, M. Kowalczyk, J. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geometric and Functional Analysis, 20 (2010), no. 4, 918-957".  Let $(M , K)$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation

$$/epsilon^2/Delta_g u/,+/, (1 - u^2)u /,=/,0/quad /mbox{in } M,$$

where $/epsilon$ is a small parameter.  Let $K/subset M$ be an $(N-1)$-dimensional smooth minimal submanifold that separates $M$ into two disjoint components. Assume that $K$ is non-degenerate in the sense that it does not support non-trivial  Jacobi fields, and that $|A_K|^2+/mbox{Ric}_g(/nu_K, /nu_K)$ is positive along $K$. 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 $K$, with mutual distance $O(/epsilon|/ln /epsilon|)$.