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: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=361893
• 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.

微分几何是研究空间，它的形状，以及两者之间的相互作用。我的研究集中在这一研究的两个关键领域。第一个涉及空间的基本结构，以及它的几何形状最终如何影响空间本身的底层结构。第二个问题是建立某些理想形状的存在。这些是具有美丽数学意义的几何物体。描述，但具体的例子是很难构建的。我用来研究这些问题的几何主要工具是里奇流方程。这个方程基本上规定了一种方法，以变形的形状，一个给定的空间到一个几何更好的一个。值得注意的是，在许多情况下，流动实际上产生了一个理想的形状，以这种方式！ 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）方式研究微分几何学称为几何分析。Ricci流是由R.S.汉密尔顿于1982年提出了一个新的数学模型，并在数学界引起了广泛的兴趣和努力。Perelman在Ricci流及其在三维几何和拓扑中的应用。里奇流在复杂几何中也有基本的应用。尽管这种兴趣越来越大，但在这个令人兴奋和肥沃的领域，加拿大的研究人员仍然相对较少。我的研究是里奇流在数学界和加拿大发展的关键组成部分。