### Abstract

We consider asymptotically flat manifolds of the form (S^{3}\{P}, G^{4}g), where G is the Green's function of the conformal Laplacian of (S^{3}, g) at a point P. We show if Ric(g) ≥ 2g and the volume of (S^{3}, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in (S^{3} \ {P},G ^{4}g). We also give an example of (S^{3}, g) where Ric(g) > 0 but (S^{3} \ {P}, G^{4}g) does have closed minimal surfaces.

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

Pages (from-to) | 395-402 |

Number of pages | 8 |

Journal | Mathematical Research Letters |

Volume | 14 |

Issue number | 2-3 |

State | Published - Mar 2007 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**A note on existence and non-existence of horizons in some asymptotically flat 3-manifolds.** / Miao, Pengzi.

Research output: Contribution to journal › Article

*Mathematical Research Letters*, vol. 14, no. 2-3, pp. 395-402.

}

TY - JOUR

T1 - A note on existence and non-existence of horizons in some asymptotically flat 3-manifolds

AU - Miao, Pengzi

PY - 2007/3

Y1 - 2007/3

N2 - We consider asymptotically flat manifolds of the form (S3\{P}, G4g), where G is the Green's function of the conformal Laplacian of (S3, g) at a point P. We show if Ric(g) ≥ 2g and the volume of (S3, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in (S3 \ {P},G 4g). We also give an example of (S3, g) where Ric(g) > 0 but (S3 \ {P}, G4g) does have closed minimal surfaces.

AB - We consider asymptotically flat manifolds of the form (S3\{P}, G4g), where G is the Green's function of the conformal Laplacian of (S3, g) at a point P. We show if Ric(g) ≥ 2g and the volume of (S3, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in (S3 \ {P},G 4g). We also give an example of (S3, g) where Ric(g) > 0 but (S3 \ {P}, G4g) does have closed minimal surfaces.

UR - http://www.scopus.com/inward/record.url?scp=34547459990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547459990&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34547459990

VL - 14

SP - 395

EP - 402

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 2-3

ER -