Canonical metrics and geometric evolutions

规范度量和几何演化

基本信息

批准号：
RGPIN-2016-03708
负责人：
Chau, Albert
金额：
$ 1.6万
依托单位：
University of British Columbia
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708701
关键词：
Canonical metrics geometric evolutions

项目摘要

Differential geometry is the study of space, its shape, and the interaction between the two. One of the most effective ways to study differential geometry is through the use of so called geometric evolution equations. In my research, I mainly focus on a geometric evolution equation known as the Ricci flow. I will now describe, in broad terms, the main goals of my research in the Ricci flow, together with their techniques and related problems. The first goal of my research is to understand what the shape of space tells us about the underlying fabric of the space. Results in this direction are often known as Uniformization Theorems in geometry. Roughly, the Ricci flow is used to deform the shape of the space to become simpler, and in so doing reveal the nature of the underlying fabric of space. One of my main results here states that when a complete Kähler manifold is positively curved in an appropriate sense, and becomes sufficiently flat at its horizon, then this space can be deformed by the Ricci flow equation to assume a flat shape, thereby identifying the nature of the underlying space. Our results have provided one of the strongest links so far in support of the Uniformization Conjecture of S.T. Yau, which states that this result is true, regardless of the behavior of curvature at the horizon. Our study of the above problem is based on an in depth study of the Ricci flow on non-compact Kähler manifolds. The second goal of my research involves the problem of determining when an underlying space can assume certain ideal shapes corresponding to geometric objects with beautiful mathematical descriptions, but for which concrete examples are very hard to construct. The complete Einstein metrics describe one such class of shapes, and a fundamental question is whether or not a given a complete Kähler manifold admits an Einstein metric. An elegant way to answer this question would be to show the given metric converges to a limit when deformed along the Ricci flow as any limit to the flow is necessarily Einstein. A closely related question is the stability of the Kähler Ricci flow at Einstein metrics. Here we consider convergence of Kähler Ricci flow starting from a metric which is a priori close to being an Einstein metric. Such results are fundamental to understanding convergence under more general hypothesis and the key is to allow for as weak a notion of the above ``closeness" as possible. A parallel goal is to study how Ricci flow deforms shapes which are poorly behaved in the sense of unbounded curvature. This extends the classical theory of Ricci flow, which demands bounded curvature, and such a study is key to addressing the above geometric problems on non-compact manifolds in full generality.

差异几何是空间的研究，两者之间的相互作用是在我的研究中使用的几何进化方程。该空间的第一个目标告诉我们有关空间的基础结构。可以假定假设的Ricci变形，从而识别迄今为止最强的链接之一歧管。确定Andunderlying空间的第二个目标可以假定具有美丽数学描述的确认形状相关性，该示例很难构造，一个基本问题是一个基本的问题，而ConpleteKähler是否承认Einstein Mentric。 o在沿着RICCI流动变形时，给定的度量收敛到限制，因为对流动的任何限制是Einstein的必要性。在更一般的假设下，关键是要尽可能弱地概念“封闭式”。一个平行的目标是研究在感知的曲率中的流动变形。