Generalization of the Ricci flow

2017 Fiscal Year Final Research Report

• Project/Area Number: 25400074
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tohoku University (2016-2017); Osaka University (2013-2015)
• Principal Investigator: Ishida Masashi 東北大学, 理学研究科, 教授 (50349023)
• Co-Investigator(Kenkyū-buntansha): 都築 正男 上智大学, 理工学部, 教授 (80296946)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: リッチフロー / 熱方程式

Outline of Final Research Achievements

We proved the differential Harnack inequality for the geometric flow in abstract setting. As a corollary, we obtained a new differential Harnack inequality for positive solutions to the conjugate heat equations associated with the Ricci flow coupled with harmonic map heat flow. We also proved a new differential Harnack inequality for the Ricci Yang-Mills flow on surface. Moreover, we introduced a reduced volume type quantity for the Ricci Yang-Mills flow on surface and proved its monotonicity under the flow. Under a mild condition on the curvature,we also proved a Perelman type no local collapsing theorem for the Ricci Yang-Mills flow in general dimension.

Free Research Field

幾何学