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Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in \(\mathbb{R}^3\). (English) Zbl 1275.53015

This is an excellent research report which contains some novel results regarding entire solutions of the Allen-Cahn equation, and their geometric meaning. The authors also provide an at-length survey on minimal surfaces \(M\) which are complete, embedded and finite total curvature, and further consider the non-degenerate ones, with \(m\geq 2\) ends. They prove that under certain circumstances, 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 a blown up surface (with ends possibly diverging logarithmically). They also prove that these solutions are non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. The construction they provide is in some sense parallel to the De Giorgi conjecture for general bounded solutions of finite Morse index.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B08 Entire solutions to PDEs
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