An Optimal Regularity Theory for Immersed Stable Minimal Hypersurfaces with Small Singular Set

TL;DR

This paper establishes optimal regularity results for immersed stable minimal hypersurfaces with small singular sets, showing they are smooth outside a set of Hausdorff dimension at most n-7.

Contribution

It proves that stable minimal hypersurfaces with singular sets of Hausdorff dimension less than n-7 are smooth outside a small singular set, providing optimal regularity conditions.

Findings

Singular set Hausdorff dimension at most n-7

Smooth minimal immersion outside the singular set

Compactness of the class under mass bounds

Abstract

We show that if $M^n$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_1(0)\setminus S$, where $S$ is closed with $\mathcal{H}^{n-2}(S)=0$, then $\text{dim}_{\mathcal{H}}\text{sing}(M)\leq n-7$, namely $\overline{M}\cap B^{n+1}_1(0)$ is represented by a smooth minimal immersion outside a closed set of generally unavoidable singularities which has Hausdorff dimension at most $n-7$. This provides the optimal a priori size assumption on the non-immersed singular set in order to guarantee optimal regularity. Consequently, such objects form a compact class under mass upper bounds.