# minimal surface

Among the surfaces $F(x,y,z)=0$, with $F$ twice continuously differentiable, a minimal surface^{} is such that in every of its points, the mean curvature^{} vanishes. Because the mean curvature is the arithmetic mean^{} of the principal curvatures^{} ${\varkappa}_{1}$ and ${\varkappa}_{2}$, the equation

$${\varkappa}_{2}=-{\varkappa}_{1}$$ |

is valid in each point of a minimal surface.

A minimal surface has also the property that every sufficiently little portion of it has smaller area than any other regular surface with the same boundary curve.

Trivially, a plane is a minimal surface. The catenoid^{} is the only surface of revolution which is also a minimal surface.

Title | minimal surface |
---|---|

Canonical name | MinimalSurface |

Date of creation | 2013-03-22 18:08:56 |

Last modified on | 2013-03-22 18:08:56 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 53A05 |

Classification | msc 26B05 |

Classification | msc 26A24 |

Related topic | PlateausProblem |

Related topic | LeastSurfaceOfRevolution |