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

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

doi:10.1007/s00039-010-0083-6

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

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

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

46 Citations (SciVal)

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 = ε_j → 0, and solutions u_ε with m-transition layers near K, with mutual distance O(ε{pipe}lnε{pipe}).

Original language	English
Pages (from-to)	918-957
Number of pages	40
Journal	Geometric and Functional Analysis
Volume	20
Issue number	4
DOIs	https://doi.org/10.1007/s00039-010-0083-6
Publication status	Published - 1 Oct 2010

Funding

Acknowledgments. The first and second authors have been supported by grants Anillo ACT 125 Center for Analysis of Partial Differential Equations (CAPDE), FONDECYT 1070389 and 1090103, and by Fondo Basal CMM-Chile. The third author is supported by an Earmarked Grant (GRF) from RGC of Hong Kong and a “Focused Research Scheme” from CUHK. The fourth author is supported by grants 10571121 and 10901108 from NSFC. He thanks the department of Mathematics of the Chinese University of Hong Kong for its kind hospitality. We thank Frank Pacard for valuable remarks.

Keywords

Concentration phenomena
multiple transition layers
positive Gauss curvature

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.1007/s00039-010-0083-6

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title = "Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature",

abstract = "Let (M,{\~g}) 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\}).",

keywords = "Concentration phenomena, multiple transition layers, positive Gauss curvature",

author = "\{del Pino\}, Manuel and Michal Kowalczyk and Juncheng Wei and Jun Yang",

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AU - del Pino, Manuel

AU - Kowalczyk, Michal

AU - Wei, Juncheng

AU - Yang, Jun

PY - 2010/10/1

Y1 - 2010/10/1

N2 - 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}).

AB - 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}).

KW - Concentration phenomena

KW - multiple transition layers

KW - positive Gauss curvature

UR - https://www.scopus.com/pages/publications/77958456826

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DO - 10.1007/s00039-010-0083-6

M3 - Article

AN - SCOPUS:77958456826

SN - 1016-443X

VL - 20

SP - 918

EP - 957

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 4

ER -

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

Abstract

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Keywords

ASJC Scopus subject areas

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