### Abstract

Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface evolves by mean curvature flow, or equivalently, that M satisfies the equation where denotes mean curvature, ν is a choice of unit normal to M, and is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature.

Original language | English |
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Pages (from-to) | 674-729 |

Number of pages | 56 |

Journal | Advances in Mathematics |

Volume | 320 |

Early online date | 14 Sep 2017 |

DOIs | |

Publication status | Published - 7 Nov 2017 |

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## Cite this

Dávila, J., Del Pino, M., & Nguyen, X. H. (2017). Finite topology self-translating surfaces for the mean curvature flow in R

^{3}.*Advances in Mathematics*,*320*, 674-729. https://doi.org/10.1016/j.aim.2017.09.014