### Abstract

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown and York (Contemporary mathematics, vol 132, American Mathematical Society, Providence, pp 129-142, 1992; Phys Rev D (3) 47(4):1407-1419, 1993) and Liu and Yau (Phys Rev Lett 90(23):231102, 2003; J Am Math Soc 19(1):181-204, 2006). Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed two dimensional surfaces evolving in an ambient three dimensional manifold. As a by-product, we are able to write the ADM mass (Arnowitt et al. in Phys. Rev. (2), 122:997-1006, 1961) of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere S_{r} and an integral of the scalar curvature plus a geometrically constructed function Φ(x) in the asymptotic region outside S_{r}. In the third part, we prove that for any closed, spacelike, 2-surface Σ in the Minkowski space R^{3} for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R^{3,1} then the Liu-Yau mass of Σ is strictly positive unless Σ lies on a hyperplane. We also show that the examples given by Ó Murchadha et al. (Phys Rev Lett 92:259001, 2004) are special cases of this result.

Original language | English (US) |
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Pages (from-to) | 437-459 |

Number of pages | 23 |

Journal | Communications in Mathematical Physics |

Volume | 298 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2010 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*298*(2), 437-459. https://doi.org/10.1007/s00220-010-1042-7