### Abstract

The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g. [2] and [10]. When this upper bound is non-positive, the second named author has shown an isoperimetric inequality for such domains, see [11]. This result again gives the comparison result for volumes alluded to above together with a characterization of the totally geodesic submanifolds of hyperbolic space forms. In the present paper we find a corresponding sharp isoperimetric inequality for minimal submanifolds in spaces with sectional curvatures bounded from above by any constant. As a corollary we find again a characterization of the totally geodesic submanifolds of spherical space forms.

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

Journal | Journal fuer die Reine und Angewandte Mathematik |

Volume | 551 |

Pages (from-to) | 101-121 |

ISSN | 0075-4102 |

Publication status | Published - 2002 |

## Cite this

Markvorsen, S., & Palmer, V. (2002). Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds.

*Journal fuer die Reine und Angewandte Mathematik*,*551*, 101-121.