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By marginally trapped surface I mean a spacelike surface in a 4-dimensional Lorentzian manifold such that the mean curvature vector is lightlike.

In my research I have stumbled across marginally trapped surfaces with Gaussian curvature equal to 1 in De Sitter's space. I have searched in vain for references about such surfaces. Does anyone know anything about them and if they could be interesting for General Relativity? I also have stumbled across spacelike surfaces with flat normal bundle in De Sitter space with constant Gaussian equal to 1, same question... Thank you in advance for any information.

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If by De Sitter you mean the manifold isometric to the exterior hyperboloid embedded in 4+1 dimensional Minkowski space: the section $t = 0$ is totally geodesic and isometric to $\mathbb{S}^3$. The equator of $\mathbb{S}^3$ is a minimal surface, and hence is a marginally trapped surface in de Sitter space. Furthermore, as the equator in $\mathbb{S}^3$ it is isometric to $\mathbb{S}^2$ and hence has constant Gaussian curvature.

The entire paragraph above can also be easily visualized from a symmetry argument.

Acting with the isometry group of De Sitter, you also get a seven-dimensional family.

They are entirely uninteresting from the point of view of general relativity: trapped and marginally trapped surfaces mainly arise as interesting objects in asymptotically flat space-times as they are indicative of black hole presence. But

  1. De Sitter is not asymptotically flat
  2. De Sitter has compact spatial section, so Penrose's incompleteness theorem cannot apply.
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  • $\begingroup$ Dear Willie Wong, thanks a lot for your answer. Yes, we are talking about "the same" De Sitter. There are many examples besides the great sphere that you have mentioned, and I'll try to see if they are interesting from a geometric point of view, since, as you have said, they are not interesting from a general relativistic point of view. $\endgroup$ – Pedro Namtior Jan 30 '16 at 11:58
  • $\begingroup$ From a pure geometry point of view: it would be kinda neat to have a full classification of all round marginally trapped surfaces in de Sitter, especially since you claimed there are ones not generated from symmetry considerations. $\endgroup$ – Willie Wong Jan 31 '16 at 18:57

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