The Kahler Ricci flow on complete non-compact Kahler manifolds and canonical Kahler metrics/structures

完全非紧卡勒流形和规范卡勒度量/结构上的卡勒里奇流

• 批准号: 327637-2006
• 负责人: Chau, Albert
• 金额: $ 0.95万
• 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: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=397885
• 关键词: Kahler; Ricci; flow; complete; non

Differential geometry is the study of space, its shape, and the interaction between the two.  My research focuses on two key areas of this study.  The first concerns the fundamental structure of space, and how its geometric shape ultimately affects the underlying fabric of the space itself.  The second is the problem of establishing the existence of certain ideal shapes.  These are geometric objects which have beautiful mathematical descriptions, but for which concrete examples are very hard to construct.  The main tool I use to study these problems in geometry is the Ricci flow equation.  This equation essentially prescribes a way to deform the shape of a given space into a geometrically nicer one.  The remarkable thing is that in many cases, the flow actually produces an ideal shape in this way!  The Ricci flow belongs to a class of equations known as geometric evolution equations, and the study of differential geomtery in this (and slightly more general) way is known as geometric analysis. The Ricci flow was first introduced by R.S. Hamilton in 1982, and has since been the focus of rapidly growing interest and efforts in the mathematical community.  This is especially due to the recent work of G. Perelman in the Ricci flow and its application to 3-dimensional geometry and topology. The Ricci flow also has fundamental applications to complex geometry. Despite this growing interest, there are still relatively few Canadian researchers in this exciting and fertile area. My research is a key component of the development of Ricci flow in both the mathematical community and in Canada.

微分几何是对空间、空间形状以及两者之间相互作用的研究。  我的研究重点是本研究的两个关键领域。  第一个涉及空间的基本结构，以及其几何形状最终如何影响空间本身的底层结构。  第二个是确定某些理想形状是否存在的问题。  这些几何对象具有美丽的数学描述，但很难构造具体的例子。  我用来研究这些几何问题的主要工具是里奇流动方程。  该方程本质上规定了一种将给定空间的形状变形为几何上更好的形状的方法。  值得注意的是，在很多情况下，流动实际上以这种方式产生了理想的形状！  里奇流属于一类称为几何演化方程的方程，以这种（稍微更一般的）方式研究微分几何被称为几何分析。 Ricci 流首先由 R.S.汉密尔顿于 1982 年提出，此后一直是数学界迅速增长的兴趣和努力的焦点。  这尤其要归功于 G. Perelman 最近在 Ricci 流及其在 3 维几何和拓扑中的应用的工作。里奇流对于复杂几何也有基本的应用。尽管人们的兴趣日益浓厚，但在这个令人兴奋且富饶的领域，加拿大研究人员仍然相对较少。我的研究是数学界和加拿大利玛窦流发展的关键组成部分。