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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=647273
• 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.*** **

差异几何形状是对空间，形状和两者之间相互作用的研究。研究差异几何形状的最有效方法之一是使用所谓的几何进化方程。在我的研究中，我主要集中于称为RICCI流的几何演化方程。现在，我将以广义的方式描述我在RICCI流程中研究的主要目标，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有关该空间的基础结构的信息。在这个方向上的结果通常称为几何学中的统一定理。粗略地，RICCI流程用于变形空间的形状，以使其变得更简单，因此可以揭示空间基础织物的性质。我这里的主要结果之一指出，当完整的kähler歧管以适当的意义积极弯曲，并且在其地平线上变得足够平坦，那么RICCI流动方程可能会变形以假定平坦的形状，从而识别基础空间的性质。我们的结果提供了迄今为止的牢固联系之一，以支持S.T.的统一化猜想。 Yau，指出，无论曲率在地平线上的行为如何，该结果都是真实的。我们对上述问题的研究是基于对非紧凑型Kähler歧管上RICCI流的深入研究。 ***我的研究的第二个目标涉及确定何时基础空间可以假设某些理想形状与具有美丽数学描述的几何对象相对应的理想形状，但是对于这些形状对应于几何形状，但对于这些几何形状对象很难构建具体的示例。完整的爱因斯坦指标描述了这样一种形状，一个基本问题是给定的kähler歧管是否承认爱因斯坦度量。回答此问题的一种优雅方法是，当沿RICCI流变形时，给定的度量收敛到限制，因为需要对流动的任何限制。一个密切相关的问题是爱因斯坦指标上KählerRicci流的稳定性。在这里，我们考虑了KählerRicci流的收敛，从预衡开始，该度量接近是爱因斯坦公制的。这样的结果对于在更一般的假设下理解融合至关重要，关键是要尽可能弱地通知上述``接近度''。 ***一个平行的目标是研究RICCI流动如何变形在无界曲率的意义上表现不佳的形状。这扩展了RICCI流的经典理论，该理论需要有界的曲率，这项研究是解决上述一般性的非compact歧管上上述几何问题的关键。**** ** **