Let us consider the following classical problem from geometry (the case \(n=3\) is basically Plateau’s problem): Let \(\Gamma\) be a smooth, closed, oriented, \((n−1\))-dimensional submanifold of \(\mathbb{R}^{n+1}\). If we consider all the smooth, compact, oriented hypersurfaces \(M \subset \mathbb{R}^{n+1}\) with \(\partial M = \Gamma\), does there exist one with least area among them?
In the case \(n+1 \le 7\) it is known that the answer is `yes’. In the case \(n+1 \ge 8\) it is known that smooth minimizers can fail to exist, but if we are fine with minimizers having singularities we can still find them and even have a certain control on their singular set: It is always of Hausdorff dimension at most \(n-7\).
For example, in the case \(n+1 = 8\) a minimizer might have singularities which are isolated points. But in this case it is actually known that a small perturbation of \(\Gamma\) suffices to get a smooth minimizer.
In a recent preprint (arXiv:2302.02253) Chodosh-Mantoulidis-Schulze extended the perturbation result to the next two dimensions, i.e. to the cases \(n+1=9\) and \(n+1=10\).
I remember vaguely that the perturbation result in the case \(n+1=8\) allowed one to extend the original proof of Schoen-Yau of the positive mass theorem in dimensions \(\le 7\) to dimension \(8\) (though I can’t find a reference for this anymore). If true, the new result could allow for a further generalization of it to dimensions \(9\) and \(10\).
edit (Feb 16th): Otis Chodosh wrote to me about the previous paragraph. The argument is as follows. One uses the `Lohkamp reduction’ to reduce the positive mass theorem to the statement that the connected sum \(T^n \# M^n\) of a torus with any closed manifold does not admit a psc-metric. And then one can run the original Schoen-Yau argument in combination with a small perturbation of the psc-metric to get rid of potential singularities in minimizing hypersurfaces.