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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

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

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.

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