Canonical metrics and geometric evolutions

微分几何是研究空间，它的形状，以及两者之间的相互作用。学习微分几何最有效的方法之一是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里奇流的几何演化方程。现在，我将概括地描述我在里奇流研究中的主要目标，以及它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空间的底层结构。在这个方向上的结果通常被称为几何上的均匀化定理。粗略地说，利玛窦流被用来改变空间的形状，使其变得更简单，从而揭示了空间底层结构的本质。我在这里的主要结论之一是，当一个完整的Kähler流形在适当的意义上是正弯曲的，并且在其视界上变得足够平坦时，那么这个空间可以通过里奇流方程来变形，从而假设一个平坦的形状，从而确定底层空间的性质。我们的研究结果提供了迄今为止支持st . t . Yau的均匀化猜想的最强有力的联系之一，该猜想指出，无论视界上的曲率如何，这个结果都是正确的。我们对上述问题的研究是基于对非紧致Kähler流形上的Ricci流的深入研究。***我研究的第二个目标涉及到一个问题，即确定一个底层空间何时可以假设某些理想形状，这些形状与具有漂亮数学描述的几何对象相对应，但具体的例子很难构建。完备爱因斯坦度规描述了一类这样的形状，一个基本问题是给定的完备Kähler流形是否允许存在爱因斯坦度规。回答这个问题的一种优雅的方法是，当给定度规沿着里奇流变形时收敛于一个极限，因为任何极限都必然是爱因斯坦流。一个密切相关的问题是Kähler里奇流在爱因斯坦度规下的稳定性。这里我们考虑Kähler Ricci流的收敛性，从一个先验的接近爱因斯坦度规的度规开始。这样的结果是在更一般的假设下理解收敛的基础，关键是允许尽可能弱的上述“接近”概念。一个平行的目标是研究里奇流如何使在无界曲率意义上表现不佳的形状变形。这是对经典Ricci流理论的扩展，该理论要求曲率有界，这是全面解决上述非紧流形几何问题的关键。* * * * *

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.*** **