{"title":"非光滑域中的伯恩斯-克兰兹刚度","authors":"Włodzimierz Zwonek","doi":"arxiv-2409.10700","DOIUrl":null,"url":null,"abstract":"Motivated by recent papers \\cite{For-Rong 2021} and \\cite{Ng-Rong 2024} we\nprove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for\nnon-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool\nin the proofs is the phenomenon of invariance of complex geodesics (and their\nleft inverses) being somehow regular at the boundary point under the mapping\nsatisfying the property as in the Burns-Krantz rigidity theorem that lets the\nproblem reduce to one dimensional problem. Additionally, we make a discussion\non bounded symmetric domains and suggest a way to prove the Burns-Krantz\nrigidity type theorem in these domains that however cannot be applied for all\nbounded symmetric domains.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"119 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Burns-Krantz rigidity in non-smooth domains\",\"authors\":\"Włodzimierz Zwonek\",\"doi\":\"arxiv-2409.10700\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Motivated by recent papers \\\\cite{For-Rong 2021} and \\\\cite{Ng-Rong 2024} we\\nprove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for\\nnon-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool\\nin the proofs is the phenomenon of invariance of complex geodesics (and their\\nleft inverses) being somehow regular at the boundary point under the mapping\\nsatisfying the property as in the Burns-Krantz rigidity theorem that lets the\\nproblem reduce to one dimensional problem. Additionally, we make a discussion\\non bounded symmetric domains and suggest a way to prove the Burns-Krantz\\nrigidity type theorem in these domains that however cannot be applied for all\\nbounded symmetric domains.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"119 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Complex Variables\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10700\",\"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 - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10700","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
受近期论文(cite{For-Rong 2021}和(cite{Ng-Rong 2024})的启发,我们证明了多圆盘和对称双圆盘非光滑边界点的边界施瓦茨定理(Burns-Krantz rigidity type theorem)。证明的基本工具是在满足伯恩斯-克兰茨刚性定理属性的映射下,复大地线(及其左反函数)在边界点处具有某种规则性,从而使问题简化为一维问题。此外,我们还讨论了有界对称域,并提出了在这些域中证明伯恩斯-克兰茨刚性定理的方法,但这一方法并不适用于所有有界对称域。
Motivated by recent papers \cite{For-Rong 2021} and \cite{Ng-Rong 2024} we
prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for
non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool
in the proofs is the phenomenon of invariance of complex geodesics (and their
left inverses) being somehow regular at the boundary point under the mapping
satisfying the property as in the Burns-Krantz rigidity theorem that lets the
problem reduce to one dimensional problem. Additionally, we make a discussion
on bounded symmetric domains and suggest a way to prove the Burns-Krantz
rigidity type theorem in these domains that however cannot be applied for all
bounded symmetric domains.