An Optimal Regularity Theory for Immersed Stable Minimal Hypersurfaces with Small Singular Set
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918486212608000 |
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| author | Minter, Paul Xiao, Zhengyi |
| author_facet | Minter, Paul Xiao, Zhengyi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05041 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Optimal Regularity Theory for Immersed Stable Minimal Hypersurfaces with Small Singular Set Minter, Paul Xiao, Zhengyi Differential Geometry Analysis of PDEs 53A10, 49Q20, 35J60 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. |
| title | An Optimal Regularity Theory for Immersed Stable Minimal Hypersurfaces with Small Singular Set |
| topic | Differential Geometry Analysis of PDEs 53A10, 49Q20, 35J60 |
| url | https://arxiv.org/abs/2605.05041 |