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Is there some result on existence of marginally trapped surfaces in spacetime 4-manifolds?

Am I right in saying that a marginal surface (like a trapped surface in general) is a compact spacelike 2-surface by definition, so if we assume a spacelike hypersurface embedded in the manifold, does that mean that there has to be a marginal surface contained in that hypersurface, or is there some other weaker existence result?

Edit: I have considered this and obviously the answer is negative, as it would depend on the spacetime as to whether it contains a marginal surface.

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  • $\begingroup$ You should post the question only on one site and wait a bit before you ask somewhere else. $\endgroup$
    – MBN
    Commented Aug 10, 2019 at 9:55
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    $\begingroup$ In general, no. The question is closely related to the question of "given an arbitrary three manifold, does there exist a (compact) embedded minimal surface". For trawling literature, I would start with Michael Eichmair's early papers and go forward/backward in the citation chain. arxiv.org/abs/1006.4601 arxiv.org/abs/1205.4301 (Eichmair is by no means the only person studying this, but he is quite knowledgeable and thorough in his citations, making his papers a good place to start.) $\endgroup$
    – Willie Wong
    Commented Aug 14, 2019 at 22:47

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