### Abstract

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let Σ be a boundary component of some compact, time-symmetric, spacelike hypersurface Ω in a time-oriented spacetime N satisfying the dominant energy condition. Suppose the induced metric on Σ has positive Gaussian curvature and all boundary components of Ω have positive mean curvature. Suppose H ≤ H_{0} where H is the mean curvature of Σ in Ω and H_{0} is the mean curvature of Σ when isometrically embedded in ℝ^{3}. If Ω is not isometric to a domain in ℝ^{3}, then the Brown-York mass of Σ in Ω is a strict local minimum of the Wang-Yau quasi-local energy of Σ.on a small perturbation of Σ in N, there exists a critical point of the Wang-Yau quasi-local energy of.

Original language | English (US) |
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Pages (from-to) | 987-1017 |

Number of pages | 31 |

Journal | Annales Henri Poincare |

Volume | 12 |

Issue number | 5 |

DOIs | |

State | Published - Jul 1 2011 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics

### Cite this

*Annales Henri Poincare*,

*12*(5), 987-1017. https://doi.org/10.1007/s00023-011-0097-0