{"title":"Metric surfaces and conformally removable sets in the plane","authors":"Dimitrios Ntalampekos","doi":"arxiv-2408.17174","DOIUrl":null,"url":null,"abstract":"We characterize conformally removable sets in the plane with the aid of the\nrecent developments in the theory of metric surfaces. We prove that a compact\nset in the plane is $S$-removable if and only if there exists a quasiconformal\nmap from the plane onto a metric surface that maps the given set to a set of\nlinear measure zero. The statement fails if we consider maps into the plane\nrather than metric surfaces. Moreover, we prove that a set is $S$-removable\n(resp. $CH$-removable) if and only if every homeomorphism from the plane onto a\nmetric surface (resp. reciprocal metric surface) that is quasiconformal in the\ncomplement of the given set is quasiconformal everywhere.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.17174","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.