### Abstract

Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω. We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.

Original language | English (US) |
---|---|

Pages (from-to) | 197-208 |

Number of pages | 12 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 1987 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Approximation by smooth embedded hypersurfaces with positive mean curvature.** / Lin, Fang-Hua.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Approximation by smooth embedded hypersurfaces with positive mean curvature

AU - Lin, Fang-Hua

PY - 1987

Y1 - 1987

N2 - Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω. We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.

AB - Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω. We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.

UR - http://www.scopus.com/inward/record.url?scp=84974308618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84974308618&partnerID=8YFLogxK

U2 - 10.1017/S0004972700026472

DO - 10.1017/S0004972700026472

M3 - Article

VL - 36

SP - 197

EP - 208

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 2

ER -