New characterizations of Ricci curvature on RCD metric measure spaces

Bang-Xian Han

doi:10.3934/dcds.2018214

Article Contents

2018, Volume 38, Issue 10: 4915-4927. Doi: 10.3934/dcds.2018214

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New characterizations of Ricci curvature on RCD metric measure spaces

Bang-Xian Han^,

Institute for applied mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

* Corresponding author: Bang-Xian Han
* Corresponding author: Bang-Xian Han

Received: September 2017

Revised: April 2018

Published: October 2018

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We prove that on a large family of metric measure spaces, if the $L^p$-gradient estimate for heat flows holds for some $p>2$, then the $L^1$-gradient estimate also holds. This result extends Savaré's result on metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on smooth metric measure spaces. As a consequence, we propose a new analysis object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci curvature of RCD space in a local way. In the proof we adopt an iteration technique based on non-smooth Bakry-Émery theory, which is a new method to study the curvature dimension condition of metric measure spaces.

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Mathematics Subject Classification: Primary: 47D07, 30L99; Secondary: 51F99.

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