{"title":"最小超曲面的双曲展开","authors":"J. Lohkamp","doi":"10.1515/agms-2018-0006","DOIUrl":null,"url":null,"abstract":"Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\\Ʃ. These are canonical conformal deformations of H\\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"6 1","pages":"128 - 96"},"PeriodicalIF":0.9000,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0006","citationCount":"8","resultStr":"{\"title\":\"Hyperbolic Unfoldings of Minimal Hypersurfaces\",\"authors\":\"J. Lohkamp\",\"doi\":\"10.1515/agms-2018-0006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\\\\Ʃ. These are canonical conformal deformations of H\\\\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.\",\"PeriodicalId\":48637,\"journal\":{\"name\":\"Analysis and Geometry in Metric Spaces\",\"volume\":\"6 1\",\"pages\":\"128 - 96\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2018-05-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1515/agms-2018-0006\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Geometry in Metric Spaces\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/agms-2018-0006\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Geometry in Metric Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/agms-2018-0006","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.
期刊介绍：
Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed.
AGMS is devoted to the publication of results on these and related topics:
Geometric inequalities in metric spaces,
Geometric measure theory and variational problems in metric spaces,
Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density,
Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds.
Geometric control theory,
Curvature in metric and length spaces,
Geometric group theory,
Harmonic Analysis. Potential theory,
Mass transportation problems,
Quasiconformal and quasiregular mappings. Quasiconformal geometry,
PDEs associated to analytic and geometric problems in metric spaces.
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