# Rigidity and flexibility phenomena in general relativity

## Abstract/Contents

- Abstract
- This thesis is devoted to the investigation of certain aspects concerning the large scale structure of asymptotically flat initial data sets for the Einstein equation in General Relativity. In the first part of this work, we show that if an asymptotically Schwarzschildean 3-manifold (M, g) of nonnegative scalar curvature contains an unbounded, properly embedded stable minimal surface, then it is isometric to the Euclidean space with its flat metric. This implies, for instance, that in presence of positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. The same result can be extended up to ambient dimension seven, under a strong stability assumption and provided the volume growth of the hypersurface in question is known a priori to be polynomial. Furthermore, we study the restrictions imposed by the dominant energy condition (DEC) on stable, marginally outer trapped surfaces. We first prove that an unbounded, stable MOTS in an initial data set (M, g, k) is conformally diffeomorphic to either the plane C or to the cylinder A (thus extending Hawking's theorem on the topology of black holes) and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly this rules out the existence of trapped regions with cylindrical boundary. If the data are asymptotically flat and have boosted harmonic asymptotics, the former case is also proven to be globally rigid in the sense that the presence of an unbounded, properly embedded stable MOTS forces an isometric embedding of (M, g, k) in the Minkowski spacetime as a space-like slice. In the second part of the thesis, which is based on joint work with Richard Schoen, we observe that the above assumptions on the data asymptotics are essentially sharp, by constructing initial data sets that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture.The gluing scheme that we adopt allows to produce a new class of N-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding:for any large $T$ we can engineer solutions where any two massive bodies do not interact at all for any time 0< t< T, in striking contrast with the Newtonian gravity scenario.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2014 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Carlotto, Alessandro | |
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Associated with | Stanford University, Department of Mathematics. | |

Primary advisor | Schoen, Richard (Richard M.) | |

Thesis advisor | Schoen, Richard (Richard M.) | |

Thesis advisor | Mazzeo, Rafe | |

Thesis advisor | Simon, L. (Leon), 1945- | |

Thesis advisor | White, Brian, 1957- | |

Advisor | Mazzeo, Rafe | |

Advisor | Simon, L. (Leon), 1945- | |

Advisor | White, Brian, 1957- |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Alessandro Carlotto. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis (Ph.D.)--Stanford University, 2014. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2014 by Alessandro Carlotto
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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