Base point estimates of the curvature tensor of the Ricci flow

Research Project

Project/Area Number	20540068
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Geometry
Research Institution	Ochanomizu University
Principal Investigator	TODA Masahito お茶の水女子大学, 大学院・人間文化創成科学研究科, 准教授 (80291566)
Project Period (FY)	2008 – 2012
Project Status	Completed (Fiscal Year 2012)
Budget Amount *help	¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000) Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000) Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000) Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000) Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000) Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords	リッチフロー / リッチ流
Research Abstract	I investigate the Ricci flow to obtain the estimate of the curvature tensor which depends only on the tensor at the base point in question and the initial data. The analysis is directly related to singularity of the Ricci flow and I clarified that the goal is almost identical to exclude the singularity modeled on the Ricci-flat manifold, depending on the initial data, as a first step. I gradually realized that the exclusion is more difficult than it appears, and requires the completely new point of view of the framework of the analytic estimate. Hence I mainly concerned with the new interpretation of the equation of the Ricci flow itself. Specifically, I setup the phase space of the metric as a suitable complexification and gave it a Kaehler structure of infinite dimension and tried to interpret the Ricci flow as a dynamical system on the phase space. So far I specified the Kaehler potential of the phase space and the Cauchy Riemann equation of the holomorphic vector field. The next goal is to understand the Ricci flow as a flow generated by suitable vector field.