Rigid circle domains with non-removable boundaries
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914959696330752 |
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| author | Rajala, Kai |
| author_facet | Rajala, Kai |
| contents | We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $Ω$ on the Riemann sphere with conformally non-removable boundary. Here rigidity means that every conformal map from $Ω$ onto another circle domain is a Möbius transformation, and non-removability means that there is a homeomorphism of the Riemann sphere which is conformal off $\partial Ω$ but not everywhere.
Our construction is based on a theorem of Wu, which states that the product of any Cantor set $E$ with a sufficiently thick Cantor set $F$ is non-removable. We show that one can choose $E$ and $F$ so that the complement of the union of $E \times F$ and suitably placed disks is rigid.
The proof of rigidity involves a metric characterization of conformal maps, which was recently proved by Ntalampekos. The other direction of the rigidity conjecture, i.e., whether removability of the boundary implies rigidity, remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigid circle domains with non-removable boundaries Rajala, Kai Complex Variables 30C20, 30C35 We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $Ω$ on the Riemann sphere with conformally non-removable boundary. Here rigidity means that every conformal map from $Ω$ onto another circle domain is a Möbius transformation, and non-removability means that there is a homeomorphism of the Riemann sphere which is conformal off $\partial Ω$ but not everywhere. Our construction is based on a theorem of Wu, which states that the product of any Cantor set $E$ with a sufficiently thick Cantor set $F$ is non-removable. We show that one can choose $E$ and $F$ so that the complement of the union of $E \times F$ and suitably placed disks is rigid. The proof of rigidity involves a metric characterization of conformal maps, which was recently proved by Ntalampekos. The other direction of the rigidity conjecture, i.e., whether removability of the boundary implies rigidity, remains open. |
| title | Rigid circle domains with non-removable boundaries |
| topic | Complex Variables 30C20, 30C35 |
| url | https://arxiv.org/abs/2409.19103 |