Quasisymmetric rectifiability of uniformly disconnected sets
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917982446288896 |
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| author | Honeycutt, Jacob Vellis, Vyron |
| author_facet | Honeycutt, Jacob Vellis, Vyron |
| contents | We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus in 1999 from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasisymmetric rectifiability of uniformly disconnected sets Honeycutt, Jacob Vellis, Vyron Metric Geometry 30L05 (Primary), 30L10, 51F99 We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus in 1999 from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively. |
| title | Quasisymmetric rectifiability of uniformly disconnected sets |
| topic | Metric Geometry 30L05 (Primary), 30L10, 51F99 |
| url | https://arxiv.org/abs/2504.08127 |