# What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

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Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent preprint, Schoen and Yau show how the usual techniques can be generalised to arbitrary dimension.

My question is: What known results can trivially said to be true in higher dimensions now, in light of this paper? Also, which related results with the same dimension restriction will be non-trivial to extend to higher dimensions?

Apologies if this question is too open-ended (this is my first time posting here) – I'm hoping it will be considered something like a "community wiki", as I think such a list would be interesting to the community.

*More precisely, see Thompson's comment below

This post imported from StackExchange MathOverflow at 2017-05-19 14:23 (UTC), posted by SE-user Steve
retagged May 19, 2017
Exciting! I would especially be interested to know if this has any bearing on the conjecture that every PSC metric on the 7-sphere extends to the 8-ball - this would kill off a lot of hard questions on the topology of the space of PSC metrics on a given manifold.

This post imported from StackExchange MathOverflow at 2017-05-19 14:23 (UTC), posted by SE-user Paul Siegel
Let me just remark that phrases like "the regularity theory for minimal hypersurfaces holds up to dimension 7" put people who work in GMT/regularity theory on edge because they are often interpreted incorrectly by people who don't work with singularities. To be careful, the relevant theorem for SY appear to be the following celebrated result: "An area-minimizing $n$-dimensional integral current in codimension 1 has a singular set (which is closed) and of dimension at most $n−7$". It's still the case that reasonably little is known about this singular set.

This post imported from StackExchange MathOverflow at 2017-05-19 14:23 (UTC), posted by SE-user Thompson
Thanks for the comment Thompson; I should have been clearer there

This post imported from StackExchange MathOverflow at 2017-05-19 14:23 (UTC), posted by SE-user Steve

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