Topological censorship from the initial data point of view

Michael Eichmair, Gregory J Galloway, Daniel Pollack

Research output: Contribution to journalArticle

13 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)389-405
Number of pages17
JournalJournal of Differential Geometry
Volume95
Issue number3
StatePublished - Nov 2013

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Null
Space-time
Theorem
Singularity
Imply
Incompleteness
Minimal surface
Cauchy
Higher Dimensions
Geodesic
Interior
Ball
Three-dimensional
Energy
Generalization

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Analysis
  • Geometry and Topology

Cite this

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

In: Journal of Differential Geometry, Vol. 95, No. 3, 11.2013, p. 389-405.

Research output: Contribution to journalArticle

Eichmair, Michael ; Galloway, Gregory J ; Pollack, Daniel. / Topological censorship from the initial data point of view. In: Journal of Differential Geometry. 2013 ; Vol. 95, No. 3. pp. 389-405.
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