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I have recently been reading some of the literature on the Penrose inequality, especially the papers by Bray and by Huisken and Ilmanen.

One notices immediately that the existing proofs for the Penrose conjecture focus on a rather special case with special symmetry for the spatial hypersurface of the spacetime: this then means that the Penrose inequality can be translated into a statement of Riemannian geometry (known as the Riemannian Penrose inequality) which can be attacked with all the machinery available in that branch of differential geometry.

Although these proofs are quite profound and make interesting uses of different geometric flows, I was wondering if there were any plausible suggestions anywhere in the literature towards actually proving the Penrose conjecture in full generality without any special symmetry assumptions? I have looked through the literature but cannot really find anything. Perhaps someone could point me to the relevant parts of the literature where some plausible suggestions have been made, perhaps also using geometric flows?

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The difficulty of a general proof was discussed in A counter-example to a recent version of the Penrose conjecture (2010): a general existence theorem cannot be expected with boundary conditions compatible with generalized apparent horizons.

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    $\begingroup$ This is a good counter-example to the 'new' version of the inequality which Bray et al. created but I suppose in general a proof might exist? $\endgroup$
    – Hollis Williams
    Jul 15, 2019 at 11:31

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