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

Abstract

We show that if $M^n$ is a properly immersed, two-sided, stable minimal hypersurface in $B_1^{n+1}(0)\setminus S$, where $S$ is closed with ${\mathcal{H}}^{n-2}(S)=0$, then ${\text{dim}}_{\mathcal{H}}\text{sing}(M)\le n-7$, namely $\overline{M}\cap B_1^{n+1}(0)$ 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.

65 pages