### Abstract

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ^{3}, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

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

Pages (from-to) | 459-475 |

Number of pages | 17 |

Journal | Inventiones Mathematicae |

Volume | 172 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2008 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*172*(3), 459-475. https://doi.org/10.1007/s00222-007-0102-x

**On the capacity of surfaces in manifolds with nonnegative scalar curvature.** / Bray, Hubert; Miao, Pengzi.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 172, no. 3, pp. 459-475. https://doi.org/10.1007/s00222-007-0102-x

}

TY - JOUR

T1 - On the capacity of surfaces in manifolds with nonnegative scalar curvature

AU - Bray, Hubert

AU - Miao, Pengzi

PY - 2008/6

Y1 - 2008/6

N2 - Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ3, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

AB - Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ3, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

UR - http://www.scopus.com/inward/record.url?scp=42449134092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42449134092&partnerID=8YFLogxK

U2 - 10.1007/s00222-007-0102-x

DO - 10.1007/s00222-007-0102-x

M3 - Article

VL - 172

SP - 459

EP - 475

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -