Metric surfaces and conformally removable sets in the plane
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910584365121536 |
|---|---|
| author | Ntalampekos, Dimitrios |
| author_facet | Ntalampekos, Dimitrios |
| contents | We characterize conformally removable sets in the plane with the aid of the recent developments in the theory of metric surfaces. We prove that a compact set in the plane is $S$-removable if and only if there exists a quasiconformal map from the plane onto a metric surface that maps the given set to a set of linear measure zero. The statement fails if we consider maps into the plane rather than metric surfaces. Moreover, we prove that a set is $S$-removable (resp. $CH$-removable) if and only if every homeomorphism from the plane onto a metric surface (resp. reciprocal metric surface) that is quasiconformal in the complement of the given set is quasiconformal everywhere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_17174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric surfaces and conformally removable sets in the plane Ntalampekos, Dimitrios Complex Variables Metric Geometry Primary 30L10, Secondary 30C35, 30C62 We characterize conformally removable sets in the plane with the aid of the recent developments in the theory of metric surfaces. We prove that a compact set in the plane is $S$-removable if and only if there exists a quasiconformal map from the plane onto a metric surface that maps the given set to a set of linear measure zero. The statement fails if we consider maps into the plane rather than metric surfaces. Moreover, we prove that a set is $S$-removable (resp. $CH$-removable) if and only if every homeomorphism from the plane onto a metric surface (resp. reciprocal metric surface) that is quasiconformal in the complement of the given set is quasiconformal everywhere. |
| title | Metric surfaces and conformally removable sets in the plane |
| topic | Complex Variables Metric Geometry Primary 30L10, Secondary 30C35, 30C62 |
| url | https://arxiv.org/abs/2408.17174 |