Entire solutions of the allen-cahn equation and complete embedded minimal surfaces of finite total curvature in r^3

Manuel del Pino, Universidad de Chile Michal Kowalczyk Universidad de Chile Juncheng Wei Chinese University of Hong Kong

Differential Geometry mathscidoc:1609.10294

Journal of Differential Geometry, 93, (1), 67-131, 2013
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 u+ f(u) = 0 in R3. Here f = −W0 with W bi-stable and balanced, for instance W(u) = 1 4 (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 > 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 := −1M, with ends possibly diverging logarithmically from M . We prove that these solutions are L1-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.
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@inproceedings{manuel2013entire,
  title={ENTIRE SOLUTIONS OF THE ALLEN-CAHN EQUATION AND COMPLETE EMBEDDED MINIMAL SURFACES OF FINITE TOTAL CURVATURE IN R^3},
  author={Manuel del Pino,, Michal Kowalczyk, and Juncheng Wei},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914073158949590956},
  booktitle={Journal of Differential Geometry},
  volume={93},
  number={1},
  pages={67-131},
  year={2013},
}
Manuel del Pino,, Michal Kowalczyk, and Juncheng Wei. ENTIRE SOLUTIONS OF THE ALLEN-CAHN EQUATION AND COMPLETE EMBEDDED MINIMAL SURFACES OF FINITE TOTAL CURVATURE IN R^3. 2013. Vol. 93. In Journal of Differential Geometry. pp.67-131. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914073158949590956.
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