Canonical metrics and geometric evolutions

规范度量和几何演化

• 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

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.

微分几何是研究空间，它的形状，以及两者之间的相互作用。 研究微分几何最有效的方法之一是通过使用所谓的几何演化方程。 在我的研究中，我主要关注一个被称为Ricci流的几何演化方程。 现在我将概括地描述我在利玛窦流研究中的主要目标，以及他们的技术和相关问题。 我研究的第一个目标是了解空间的形状告诉我们关于空间的基本结构。 在这个方向上的结果通常被称为几何中的一致化定理。 大致上，里奇流被用来变形空间的形状，使其变得更简单，并在这样做的过程中揭示了空间底层结构的本质。 我在这里的一个主要结果表明，当一个完备的凯勒流形在适当的意义上是正弯曲的，并且在其视界处变得足够平坦时，这个空间可以被里奇流方程变形为平坦的形状，从而确定底层空间的性质。 我们的结果提供了一个强有力的链接，到目前为止，在支持的一致化猜想的S. T。它指出这个结果是正确的，不管视界处曲率的行为如何。 我们对上述问题的研究是基于对非紧Kähler流形上Ricci流的深入研究。 我研究的第二个目标是确定一个底层空间何时可以呈现出某些理想的形状，这些形状对应于具有美丽数学描述的几何对象，但很难构建具体的例子。 完备的爱因斯坦度量描述了一类这样的形状，一个基本的问题是给定的完备凯勒流形是否允许爱因斯坦度量。 回答这个问题的一个优雅的方法是表明，当给定的度规沿着里奇流沿着变形时，会收敛到一个极限，因为该流的任何极限都必然是爱因斯坦的。 一个密切相关的问题是在爱因斯坦度规下凯勒里奇流的稳定性。 在这里，我们考虑收敛的凯勒利玛窦流开始从一个度量，这是一个先验接近爱因斯坦度量。 这样的结果对于理解在更一般的假设下的收敛性是至关重要的，关键是允许尽可能弱的上述"接近”的概念。 一个平行的目标是研究如何里奇流变形的形状，表现不佳的意义上的无界曲率。 这扩展了经典的Ricci流理论，它要求有界曲率，这样的研究是解决上述非紧流形上的几何问题的关键。