{"title":"The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces","authors":"Karl-Theodor Sturm","doi":"10.1090/memo/1443","DOIUrl":null,"url":null,"abstract":"Equipped with the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 2 comma q\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^{2,q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-distortion distance <inline-formula content-type=\"math/tex\"> <tex-math> \\DD _{2,q}</tex-math></inline-formula>, the space <inline-formula content-type=\"math/tex\"> <tex-math> \\XX _{2q}</tex-math></inline-formula> of all metric measure spaces <inline-formula content-type=\"math/tex\"> <tex-math> (X,\\d ,\\m )</tex-math></inline-formula> is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on <inline-formula content-type=\"math/tex\"> <tex-math> \\ol \\XX _{2q}</tex-math></inline-formula> are presented.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"78","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/memo/1443","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 78
Abstract
Equipped with the L2,qL^{2,q}-distortion distance \DD _{2,q}, the space \XX _{2q} of all metric measure spaces (X,\d ,\m ) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on \ol \XX _{2q} are presented.
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