# Regularity of minimizing hypersurfaces

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.