Geometry of metric measure spaces and optimal transport theory

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740048
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: OHTA Shin-ichi 京都大学, 理学(系)研究科(研究院), 准教授 (00372558)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: リーマン幾何 / 曲率 / 熱流 / フィンスラー幾何 / 勾配流

Outline of Final Research Achievements

We consider two generalizations of the curvature-dimension condition, which is a notion of a lower Ricci curvature bound for metric measure spaces. One generalization is to make the parameter regarded as an “upper bound of the dimension” being negative. Then the condition is getting weaker and covers a wider class of spaces.
We also establish the Bochner formula for Finsler manifolds by using the weighted Ricci curvature. As applications, we obtain some gradient estimates for heat flow, and a generalization of Cheeger-Gromoll’s classical splitting theorem.

Free Research Field

微分幾何学