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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

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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Apr 29, 2017 at 11:03
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    $\begingroup$ 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. $\endgroup$
    – Thompson
    Commented Apr 30, 2017 at 14:09
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    $\begingroup$ Thanks for the comment Thompson; I should have been clearer there $\endgroup$
    – Steve McCormick
    Commented May 3, 2017 at 8:54

3 Answers 3

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I know two applications:
a) The convergence of the Yamabe flow in dimension 6 and higher
Simon Brendle, Invent. math. 170, 541–576 (2007)
DOI: 10.1007/s00222-007-0074-x
b) Solution of the equivariant Yamabe problem
Farid Madani, Hebey-Vaugon conjecture II. (English, French summary)
C. R. Math. Acad. Sci. Paris 350 (2012), no. 17-18, 849–852.

Both article use the positive mass theorem to construct a good test function for the Yamabe functional.

For application b) the history is a bit distorted and not easily visible, so I will give more explanation.

Let $G$ be a compact Lie group acting on a compact manifold $M$. The equivariant Yamabe problem is to show the following. If $g_0$ is a $G$-invariant Riemannian metric on $M$, let $[g_0]^G$ be the set of all $G$-invariant metrics of volume 1 conformal to $g_0$.

The equivariant Yamabe problem is to minimize the Einstein-Hilbert functional $$g\mapsto \int_M~ \mathrm{Scal}^g~ dv^g$$ among all metrics $g\in [g_0]$ with the constraint $\mathrm{vol}(M,g)=1$. It was considered as being solved by an article
E. Hebey, M. Vaugon, Le problème de Yamabe équivariant, Bull. Sci. Math. 117 (1993) 241–286.
however this article used Schoen's Weyl vanishing conjecture which turned out not to hold in large dimensions (Counterexample by Brendle). Madani had developed in the above reference an alternative proof, closer to the standard proof of the classical Yamabe problem. In some cases he uses the positive mass theorem to construct a good test function. With the positive mass theorem in general, the solution of the equivariant Yamabe problem follows in full generality.

Probably Brendle's test function from the publication in Inventiones Mathematicae cited above could have been used for the same purpose, but I do not know a reference where it was worked out how to use Brendle's test function to solve the equivariant Yamabe problem.

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The progress of Schoen and Yau also allows to generalize many statements previously known only for spin manifolds to non-spin manifolds. See e.g. a preprint by Thomas Schick and Simone Cecchini, arXiv math.GT 1810.02116. In particular, they are now able to prove the following: An enlargeable metric (complete or not) cannot have uniformly positive scalar curvature. Therefore, a closed enlargeable manifold cannot carry any metric of positive scalar curvature.

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The compactness of solutions to the Yamabe problem (and its version with boundary) holds upto dimension 24 if the PMT is true. (For higher dimensions there are counterexamples.)

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