{"title":"平面中的公设曲面和保角可移集","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":"{\"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}","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}
Metric surfaces and conformally removable sets in the plane
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.