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](https://arxiv.org/abs/1704.05490), Schoen and Yau show how the usual techniques can be generalised to arbitrary dimension.



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> 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?
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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



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


