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**On the Riemannian Penrose inequality in dimensions less than eight.**
*(English)*
Zbl 1168.53016

Authors’ abstract: The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal hypersurface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, Bray extended this result to the general case of multiple black holes using a different technique. In this article, we extend the technique of to dimensions less than eight. Part of the argument is contained in a companion article by Lee. The equality case of the theorem requires the added assumption that the manifold be spin.

Reviewer: Gil de Oliveira-Neto (Minas Gerais)

### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C80 | Applications of global differential geometry to the sciences |

83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |

83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |

83C57 | Black holes |

83E15 | Kaluza-Klein and other higher-dimensional theories |

### Keywords:

Riemannian Penrose inequality; asymptotically flat manifolds; ADM mass; black holes; dimensions less than eight; nonnegative scalar curvature
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\textit{H. L. Bray} and \textit{D. A. Lee}, Duke Math. J. 148, No. 1, 81--106 (2009; Zbl 1168.53016)

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