### Abstract

We 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 in ℝ^{3}. Here f = −W′ with W bi-stable and balanced, for instance W(u) = 1/4(1 - u^{2})^{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 M_{α}: = α^{−} M, with ends possibly diverging logarithmically from M_{α}. We prove that these solutions are L^{∞}-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.

Original language | English |
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Pages (from-to) | 67-131 |

Number of pages | 65 |

Journal | Journal of Differential Geometry |

Volume | 93 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2013 |

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

## Cite this

^{3}.

*Journal of Differential Geometry*,

*93*(1), 67-131. https://doi.org/10.4310/jdg/1357141507