Open Access
November 2013 Topological censorship from the initial data point of view
Michael Eichmair, Gregory J. Galloway, Daniel Pollack
J. Differential Geom. 95(3): 389-405 (November 2013). DOI: 10.4310/jdg/1381931733


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 $\mathbb{R}^3$. 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-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to $\mathbb{R}^3$ 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 $\mathbb{R}^3$ minus a finite number of open balls. An extension to higher dimensions is also discussed.


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Michael Eichmair. Gregory J. Galloway. Daniel Pollack. "Topological censorship from the initial data point of view." J. Differential Geom. 95 (3) 389 - 405, November 2013.


Published: November 2013
First available in Project Euclid: 16 October 2013

zbMATH: 1279.83027
MathSciNet: MR3128989
Digital Object Identifier: 10.4310/jdg/1381931733

Rights: Copyright © 2013 Lehigh University

Vol.95 • No. 3 • November 2013
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