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Base point estimates of the curvature tensor of the Ricci flow

里奇流曲率张量的基点估计

基本信息

批准号：
20540068
负责人：
TODA Masahito
金额：
$ 2.33万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2008
资助国家：
日本
起止时间：
2008 至 2012
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-20540068/
关键词：
リッチフローリッチ流

项目摘要

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.

我研究里奇流，以获得曲率张量的估计，它只取决于所讨论的基点和初始数据的张量。分析与里奇流的奇点直接相关，我澄清说，根据初始数据，作为第一步，我们的目标几乎是相同的，即排除里奇平坦流形上的奇点。我逐渐意识到，排除比它看起来要困难得多，它需要分析估计框架的全新观点。因此，我主要关心的是对里奇流方程本身的新解释。具体而言，我将度规的相空间设置为合适的复化形式，并赋予其无限维的Kaehler结构，并试图将Ricci流解释为相空间上的动力系统。到目前为止，我已经说明了相空间的凯勒势和全纯向量场的柯西黎曼方程。下一个目标是将里奇流理解为由合适的矢量场产生的流。