### Abstract

Inspired by the work of Chen-Zhang [5], we derive an evolution formula for theWang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau [4]. If the reference static space represents a mass minimizing, static extension of the initial surface ∑, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasilocal mass at ∑. Combining the evolution formula for the quasi-local energy with a localized Penrose inequality proved in [9], we prove a rigidity theorem for compact 3-manifolds with nonnegative scalar curvature, with boundary. This rigidity theorem in turn gives a characterization of the equality case of the localized Penrose inequality in 3-dimension.

Original language | English (US) |
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Pages (from-to) | 1411-1426 |

Number of pages | 16 |

Journal | Advances in Theoretical and Mathematical Physics |

Volume | 23 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2019 |

### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)

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## Cite this

*Advances in Theoretical and Mathematical Physics*,

*23*(5), 1411-1426. https://doi.org/10.4310/ATMP.2019.v23.n5.a5