{"title":"On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds","authors":"Vitali Kapovitch, Xingyu Zhu","doi":"10.1007/s10455-023-09920-1","DOIUrl":null,"url":null,"abstract":"<div><p>We show that if an Alexandrov space <i>X</i> has an Alexandrov subspace <span>\\({\\bar{\\Omega }}\\)</span> of the same dimension disjoint from the boundary of <i>X</i>, then the topological boundary of <span>\\({\\bar{\\Omega }}\\)</span> coincides with its Alexandrov boundary. Similarly, if a noncollapsed <span>\\({{\\,\\textrm{RCD}\\,}}(K,N)\\)</span> space <i>X</i> has a noncollapsed <span>\\({{\\,\\textrm{RCD}\\,}}(K,N)\\)</span> subspace <span>\\({\\bar{\\Omega }}\\)</span> disjoint from boundary of <i>X</i> and with mild boundary condition, then the topological boundary of <span>\\({\\bar{\\Omega }}\\)</span> coincides with its De Philippis–Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.\n</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 2","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Global Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10455-023-09920-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We show that if an Alexandrov space X has an Alexandrov subspace \({\bar{\Omega }}\) of the same dimension disjoint from the boundary of X, then the topological boundary of \({\bar{\Omega }}\) coincides with its Alexandrov boundary. Similarly, if a noncollapsed \({{\,\textrm{RCD}\,}}(K,N)\) space X has a noncollapsed \({{\,\textrm{RCD}\,}}(K,N)\) subspace \({\bar{\Omega }}\) disjoint from boundary of X and with mild boundary condition, then the topological boundary of \({\bar{\Omega }}\) coincides with its De Philippis–Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.
期刊介绍:
This journal examines global problems of geometry and analysis as well as the interactions between these fields and their application to problems of theoretical physics. It contributes to an enlargement of the international exchange of research results in the field.
The areas covered in Annals of Global Analysis and Geometry include: global analysis, differential geometry, complex manifolds and related results from complex analysis and algebraic geometry, Lie groups, Lie transformation groups and harmonic analysis, variational calculus, applications of differential geometry and global analysis to problems of theoretical physics.