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

Manuel del Pino, Michal Kowalczyk, Juncheng Wei, Jun Yang

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)

Abstract

Let (M,g̃) be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation where ε is a small parameter. Let K ⊂ M be an (N - 1)-dimensional smooth minimal submanifold that separates M into two disjoint components. Assume that K is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that is positive along K. Then for each integer m ≥ 2, we establish the existence of a sequence ε = εj → 0, and solutions uε with m-transition layers near K, with mutual distance O(ε{pipe}lnε{pipe}).

Original languageEnglish
Pages (from-to)918-957
Number of pages40
JournalGeometric and Functional Analysis
Volume20
Issue number4
DOIs
Publication statusPublished - 1 Oct 2010

Keywords

  • Concentration phenomena
  • multiple transition layers
  • positive Gauss curvature

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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