Topological censorship from the initial data point of view

TL;DR

This paper generalizes topological censorship and singularity theorems using initial data concepts, showing that certain trapped surfaces imply spacetime incompleteness or specific topological structures.

Contribution

It introduces immersed marginally outer trapped surfaces and proves their significance in initial data sets, extending classical theorems to a broader initial data context.

Findings

Presence of immersed marginally outer trapped surfaces implies null geodesic incompleteness.

Initial data sets without such surfaces are topologically simple, diffeomorphic to R^3 minus finite balls.

Results extend to higher dimensions under certain conditions.

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-time-symmetric version of a theorem of Meeks-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.