Characterization of quasispheres via smooth approximation
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910827469078528 |
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| author | Ntalampekos, Dimitrios |
| author_facet | Ntalampekos, Dimitrios |
| contents | We prove that every two-dimensional quasisphere is the limit of a sequence of smooth spheres that are uniform quasispheres. In the case of metric spheres of finite area we provide necessary and sufficient geometric conditions for a quasisphere, involving the doubling property, linear local connectivity, the Loewner property, conformal modulus, and reciprocity. In particular, although an arbitrary quasisphere does not satisfy necessarily all of those geometric conditions, we prove that every quasisphere can be approximated by uniform quasispheres that are uniformly doubling, linearly locally connected, 2-Loewner, reciprocal and satisfy a uniform bound for the modulus of annuli. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10006 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of quasispheres via smooth approximation Ntalampekos, Dimitrios Metric Geometry Complex Variables Geometric Topology Primary 30L10, 30C65, 53C23, Secondary 30F10, 51F99, 53A05 We prove that every two-dimensional quasisphere is the limit of a sequence of smooth spheres that are uniform quasispheres. In the case of metric spheres of finite area we provide necessary and sufficient geometric conditions for a quasisphere, involving the doubling property, linear local connectivity, the Loewner property, conformal modulus, and reciprocity. In particular, although an arbitrary quasisphere does not satisfy necessarily all of those geometric conditions, we prove that every quasisphere can be approximated by uniform quasispheres that are uniformly doubling, linearly locally connected, 2-Loewner, reciprocal and satisfy a uniform bound for the modulus of annuli. |
| title | Characterization of quasispheres via smooth approximation |
| topic | Metric Geometry Complex Variables Geometric Topology Primary 30L10, 30C65, 53C23, Secondary 30F10, 51F99, 53A05 |
| url | https://arxiv.org/abs/2502.10006 |