Research demonstrates that area-minimizing hypersurfaces are generally smooth in dimension 11, suggesting implications for the Plateau problem and singular sets.
We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension $11$ in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, n+1 ≥ 12, we prove in the same two contexts that area-minimizing hypersurfaces have at most an n-10-εₙ dimensional singular set after an arbitrarily C^∞-small perturbation of the Plateau boundary or the ambient Riemannian metric, respectively.
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Chodosh et al. (2025) studied this question.
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