On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces

• 成果类型: Article
• 署名作者: Erbar, Matthias; Kuwada, Kazumasa; Sturm, Karl-Theodor
• 署名单位: University of Bonn; Institute of Science Tokyo; Tokyo Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0563-7
• 发表日期: 2015
• 页码: 993-1071
• 关键词: ricci curvature; li-yau; eulerian calculus; heat-flow; alexandrov; contraction; geometry

摘要

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry-A parts per thousand mery (via energy and -calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the -Wasserstein distance.