Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives

This2001introductiontreatstheclassicalisoperimetricinequalityinEuclideanspaceandcontrastingroughinequalitiesinnoncompactRiemannianmanifolds.InEuclideanspacetheemphasisisonamostgeneralformoftheinequalitysufficientlyprecisetocharacterizethecaseofequality,andinRiemannianmanifoldstheemphasisisonthosequalitativefeaturesoftheinequalitywhichprovideinsightintothecoarsegeometryatinfinityofRiemannianmanifolds.ThetreatmentinEuclideanspacefeaturesanumberofproofsoftheclassicalinequalityinincreasinggenerality,providingintheprocessatransitionfromthemethodsofclassicaldifferentialgeometrytothoseofmoderngeometricmeasuretheory;andthetreatmentinRiemannianmanifoldsfeaturesdiscretizationtechniques,andapplicationstoupperboundsoflargetimeheatdiffusioninRiemannianmanifolds.Theresultisanintroductiontotherichtapestryofideasandtechniquesofisoperimetricinequalities.