### Abstract

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi–Tam and Wang–Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown–York quasi-local mass without assuming that the boundary 2-sphere has positive Gauss curvature.

Original language | English (US) |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

DOIs | |

State | Accepted/In press - Oct 14 2016 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Mantoulidis, C., & Miao, P. (Accepted/In press). Total Mean Curvature, Scalar Curvature, and a Variational Analog of Brown–York Mass.

*Communications in Mathematical Physics*, 1-16. https://doi.org/10.1007/s00220-016-2767-8