### Abstract

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to ℝ^{3}. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together, these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-timesymmetric version of a theorem ofMeeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to ℝ^{3} contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to ℝ^{3} minus a finite number of open balls. An extension to higher dimensions is also discussed.

Original language | English (US) |
---|---|

Pages (from-to) | 389-405 |

Number of pages | 17 |

Journal | Journal of Differential Geometry |

Volume | 95 |

Issue number | 3 |

State | Published - Nov 2013 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology

### Cite this

*Journal of Differential Geometry*,

*95*(3), 389-405.

**Topological censorship from the initial data point of view.** / Eichmair, Michael; Galloway, Gregory J; Pollack, Daniel.

Research output: Contribution to journal › Article

*Journal of Differential Geometry*, vol. 95, no. 3, pp. 389-405.

}

TY - JOUR

T1 - Topological censorship from the initial data point of view

AU - Eichmair, Michael

AU - Galloway, Gregory J

AU - Pollack, Daniel

PY - 2013/11

Y1 - 2013/11

N2 - We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to ℝ3. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together, these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-timesymmetric version of a theorem ofMeeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to ℝ3 contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to ℝ3 minus a finite number of open balls. An extension to higher dimensions is also discussed.

AB - We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to ℝ3. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together, these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-timesymmetric version of a theorem ofMeeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to ℝ3 contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to ℝ3 minus a finite number of open balls. An extension to higher dimensions is also discussed.

UR - http://www.scopus.com/inward/record.url?scp=84888106423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888106423&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84888106423

VL - 95

SP - 389

EP - 405

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 3

ER -