## Abstract

We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair et al. (Commun Math Phys 386(1):253–268, 2021) in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions.

Original language | English (US) |
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Pages (from-to) | 451-480 |

Number of pages | 30 |

Journal | Communications in Mathematical Physics |

Volume | 396 |

Issue number | 2 |

DOIs | |

State | Published - Dec 2022 |

### Bibliographical note

Funding Information:A. Alaee acknowledges the support of an AMS-Simons travel grant. M. Khuri acknowledges the support of NSF Grants DMS-1708798, DMS-2104229, and Simons Foundation Fellowship 681443.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.