2009/02/12 by Manuel del Pino, del Pino, Manuel, Mike Kowalczyk +3 · 2 citations
Mathematics · #35B33 #35B40 #35J20 #35J25 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35B33 #msc:35B40 #msc:35J20 #msc:35J25
paper · pdf · doi:10.48550/arxiv.0902.2047
73pages
arxiv created 2009/02/12 · openalex publication_date 2009/02/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider minimal surfaces M which are complete, embedded and have finite total curvature in \R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation Δu + f(u) = 0 \hboxin \R3 . Here f=-W' with W bistable and balanced, for instance W(u) =\frac 14 (1-u2)2. We assume that M has m≥ 2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman-Meeks surface of any genus). We prove that for any small α>0, the Allen-Cahn equation has a family of bounded solutions depending on m-1 parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface Mα:= α-1 M, with ends possibly diverging logarithmically from M_\A. We prove that these solutions are L^∞-\em non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.