Curvature-Dimension Bounds and Functional InequalitiesLocalization, Tensorization and Stability
Curvature-Dimension Bounds and Functional Inequalities
Localization, Tensorization and Stability
dc.contributor.advisor | Sturm, Karl-Theodor | |
dc.contributor.author | Bacher, Kathrin | |
dc.date.accessioned | 2020-04-15T12:57:46Z | |
dc.date.available | 2020-04-15T12:57:46Z | |
dc.date.issued | 16.03.2010 | |
dc.identifier.uri | https://hdl.handle.net/20.500.11811/4544 | |
dc.description.abstract | This work is devoted to the analysis of abstract metric measure spaces (M,d,m) satisfying the curvature-dimension condition CD(K,N) presented by Sturm and in a similar form by Lott and Villani. In the first part, we introduce the notion of a Borell-Brascamp-Lieb inequality in the setting of metric measure spaces denoted by BBL(K,N). This inequality holds true on metric measure spaces fulfilling the curvature-dimension condition CD(K,N) and is stable under convergence of metric measure spaces with respect to the transportation distance. In the second part, we prove that the local version of CD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the usual global one. This so-called reduced curvature-dimension condition CD*(K,N) has the localization property. Furthermore, we show its stability and the tensorization property. As an application we conclude that the fundamental group of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N. In the third part, we study cones over metric measure spaces. We deduce that the n-Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the n-spherical cone over the same manifold fulfills CD(n,n+1). | en |
dc.language.iso | eng | |
dc.rights | In Copyright | |
dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | |
dc.subject | Geometrische Analysis | |
dc.subject | Differentialgeometrie | |
dc.subject | metrische Maßräume | |
dc.subject | verallgemeinerte Krümmungs-Dimensionsschranken | |
dc.subject | Funktionale Ungleichungen | |
dc.subject | Borell-Brascamp-Lieb-Ungleichung | |
dc.subject | Stabilität von Krümmungs-Dimensionsschranken | |
dc.subject | Lokalisierung | |
dc.subject | Tensorisierung | |
dc.subject | Euklidische Kegel | |
dc.subject | Sphärische Kegel | |
dc.subject | Riemannsche Mannigfaltigkeiten | |
dc.subject | geometric Analysis | |
dc.subject | differential geometry | |
dc.subject | metric measure spaces | |
dc.subject | generalized curvature-dimension bounds | |
dc.subject | functional inequalities | |
dc.subject | Borell-Brascamp-Lieb inequality | |
dc.subject | stability of curvature-dimension bounds | |
dc.subject | localization | |
dc.subject | tensorization | |
dc.subject | Euclidean cones | |
dc.subject | spherical cones | |
dc.subject | Riemannian manifolds | |
dc.subject.ddc | 510 Mathematik | |
dc.title | Curvature-Dimension Bounds and Functional Inequalities | |
dc.title.alternative | Localization, Tensorization and Stability | |
dc.type | Dissertation oder Habilitation | |
dc.publisher.name | Universitäts- und Landesbibliothek Bonn | |
dc.publisher.location | Bonn | |
dc.rights.accessRights | openAccess | |
dc.identifier.urn | https://nbn-resolving.org/urn:nbn:de:hbz:5N-20646 | |
ulbbn.pubtype | Erstveröffentlichung | |
ulbbnediss.affiliation.name | Rheinische Friedrich-Wilhelms-Universität Bonn | |
ulbbnediss.affiliation.location | Bonn | |
ulbbnediss.thesis.level | Dissertation | |
ulbbnediss.dissID | 2064 | |
ulbbnediss.date.accepted | 05.03.2010 | |
ulbbnediss.fakultaet | Mathematisch-Naturwissenschaftliche Fakultät | |
dc.contributor.coReferee | Thalmaier, Anton |
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