Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$

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dc.creatordel Pino, Manuel
dc.creatorKowalczyk, Mike
dc.creatorWei, Juncheng
dc.date2009-02-12
dc.date.accessioned2026-07-07T12:40:54Z
dc.date.available2026-07-07T12:40:54Z
dc.descriptionWe consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $Δu + f(u) = 0 \hbox{in} \R^3 $. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that $M$ has $m\ge 2$ 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 $M_α:= α^{-1} M$, with ends possibly diverging logarithmically from $M_\A$. We prove that these solutions are $L^\infty$-{\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.
dc.description73pages
dc.identifierhttps://arxiv.org/abs/0902.2047
dc.identifierhttp://arxiv.org/abs/0902.2047
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/219593
dc.subjectAnalysis of PDEs
dc.subjectDifferential Geometry
dc.subject35J25, 35J20, 35B33, 35B40
dc.titleEntire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$
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