Curvature bounds and heat kernels: discrete versus continuous spaces [Elektronische Ressource] / vorgelegt von Anca-Iuliana Bonciocat

Curvature bounds and heatkernels: discrete versus continuousspacesDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematish-Naturwissenschaftlichen Fakult¨atderRheinischen Friedrich-Wilhelms-Universit¨at Bonnvorgelegt vonAnca-Iuliana BonciocatausSlatina, Rum¨anienBonn 2008Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Facult atder Rheinischen Friedrich-Wilhelms-Universit at Bonn1. Gutachter: Prof. Dr. Karl-Theodor Sturm2.hter: Prof. Dr. Lucian BezneaTag der Promotion: 12 Juli 2008AbstractWe introduce and study rough (approximate) lower curvature bounds and roughcurvature-dimension conditions for discrete spaces and for graphs. These notionsextend the ones introduced in [St06a] and [St06b] to a larger class of non-geodesicmetric measure spaces. They are stable under an appropriate notion of convergencein the sense that the metric measure space which is approximated by a sequence ofdiscrete spaces with rough curvature≥K will have curvature≥K in the sense of[St06a]. Moreover,intheconversedirection,discretizationsofmetricmeasurespaceswithcurvature≥K willhaveroughcurvature≥K. Weapplyourresultstoconcreteexamples of homogeneous planar graphs. We derive perturbed transportation costinequalities,thatimplymassconcentrationandexponentialintegrabilityofLipschitzmaps. For spaces that satisfy a rough curvature-dimension condition we prove ageneralized Brunn-Minkowski inequality and a Bonnet-Myers type theorem.