### Abstract

The nonlocal s-fractional minimal surface equation for Σ = ∂E where E is an open set in R
^{N} is given by H
_{Σ}
^{s} (p):=
_{RN}
^{χE(}x) − χEc(x
^{)} dx = 0 for all p ∈ Σ. |x − p|
^{N}+s Here 0 < s < 1, χ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting s → 1. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal s−minimal surfaces. When s is close to 1, we first construct a connected embedded s-minimal surface of revolution in R
^{3}, the nonlocal catenoid, an analog of the standard catenoid |x
_{3}| = log(r+ r
^{2} − 1). Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone |x
_{3}| = r1 − s. We also find a two-sheet embedded s-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes. On the other hand, for any 0 < s < 1, n, m ≥ 1, s−minimal Lawson cones |v| = α|u|, (u, v) ∈ R
^{n} × R
^{m}, are found to exist. In sharp contrast with the classical case, we prove their stability for small s and n + m = 7, which suggests that unlike the classical theory (or the case s close to 1), the regularity of s-area minimizing surfaces may not hold true in dimension 7.

Original language | English |
---|---|

Pages (from-to) | 111-175 |

Number of pages | 65 |

Journal | The Journal of Differential Geometry |

Volume | 109 |

Issue number | 1 |

Early online date | 4 May 2018 |

DOIs | |

Publication status | Published - 31 May 2018 |

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*The Journal of Differential Geometry*,

*109*(1), 111-175. https://doi.org/10.4310/jdg/1525399218