Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry

2019/06/13 by Rayssa Caju, Caju, Rayssa, Pedro Dinis Gaspar +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1906.05938

openalex publication_date 2019/06/13 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

We prove that given a minimal hypersurface Γ in a compact Riemannian manifold M without boundary, if all the Jacobi fields of Γ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation -ε²Δu +W'(u)=0 on M, for sufficiently small ε>0, whose nodal sets converge to Γ. This extends the results of Pacard-Ritoré (in the case of closed manifolds and zero mean curvature).

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