2015/09/24 by Luigi Ambrosio, Nicola Gigli, Andrea Mondino +2
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Applied mathematics #Bounded function #Computer science #Context (archaeology) #Convexity #Curvature #Curvature of Riemannian manifolds #Differential (mechanical device) #Differential geometry #Dimension (graph theory) #Geometric Analysis and Curvature Flows #Geometric flow #Geometry #Geometry and complex manifolds #Hessian matrix #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Measure (data warehouse) #Metric (unit) #Metric space #Nonlinear Partial Differential Equations #Order (exchange) #Physics #Point processes and geometric inequalities #Pure mathematics #Ricci curvature #Ricci decomposition #Ricci flow #Riemann curvature tensor #Scalar curvature #Sectional curvature #Semigroup #math.AP #math.FA #math.MG #math.PR #msc:30L99 #msc:47D07 #msc:49Q20
paper · pdf · doi:10.1090/memo/1196
published as Mem. Amer. Math. Soc. 262 (2019), no. 1270, v+121 pp · 115 pages. Minor typos corrected and many small improvements added. Thm. 3.4, Lemma 5.4, Thm. 9.15, Thm. 9.21, Lemma 12.2, Lemma 12.4, Thm. 12.7, Thm. 12.8, Lemma 12.11 augmented/reenforced. Thm. 3.6, Sec. 5.6, Lemma 5.5 added
openalex publication_date 2015/09/24 · openalex created_date 2016/06/24 · arxiv created 2017/02/10 · arxiv updated 2020/02/12 · openalex updated_date 2026/08/05
Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD*(K,N) condition of Bacher-Sturm.