Structure of two-dimensional mod$(q)$ area-minimizing currents near flat singularities: the codimension one case
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908416086114304 |
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| author | Skorobogatova, Anna Spolaor, Luca Stuvard, Salvatore |
| author_facet | Skorobogatova, Anna Spolaor, Luca Stuvard, Salvatore |
| contents | We obtain a fine structural result for two-dimensional mod$(q)$ area-minimizing currents of codimension one, close to flat singularities. Precisely, we show that, locally around any such singularity, the current is a $C^{1,α}$-perturbation of the graph of a radially homogeneous special multiple-valued function that arises from a superposition of homogeneous harmonic polynomials. Additionally, as a preliminary step towards an analogous result in arbitrary codimension, we prove in general that the set of flat singularities of density $\frac{q}{2}$, where the current is ``genuinely mod$(q)$", consists of isolated points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure of two-dimensional mod$(q)$ area-minimizing currents near flat singularities: the codimension one case Skorobogatova, Anna Spolaor, Luca Stuvard, Salvatore Analysis of PDEs Differential Geometry 49Q15, 49Q20, 49Q05, 53A10 We obtain a fine structural result for two-dimensional mod$(q)$ area-minimizing currents of codimension one, close to flat singularities. Precisely, we show that, locally around any such singularity, the current is a $C^{1,α}$-perturbation of the graph of a radially homogeneous special multiple-valued function that arises from a superposition of homogeneous harmonic polynomials. Additionally, as a preliminary step towards an analogous result in arbitrary codimension, we prove in general that the set of flat singularities of density $\frac{q}{2}$, where the current is ``genuinely mod$(q)$", consists of isolated points. |
| title | Structure of two-dimensional mod$(q)$ area-minimizing currents near flat singularities: the codimension one case |
| topic | Analysis of PDEs Differential Geometry 49Q15, 49Q20, 49Q05, 53A10 |
| url | https://arxiv.org/abs/2506.17813 |