Canonical metrics and geometric evolutions

微分几何是研究空间、空间形状以及两者之间相互作用的学科。研究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Ricci流的几何演化方程。现在，我将概括地描述我在利玛窦流动研究中的主要目标，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统一定理。粗略地说，利玛窦流动被用来使空间的形状变得更简单，并通过这样做来揭示空间底层结构的本质。这里我的一个主要结果是，当一个完备的Kähler流形在适当的意义上是正曲线的，并且在它的视界变得足够平坦时，那么这个空间可以通过Ricci流方程变形为平坦的形状，从而识别下面空间的性质。我们的结果提供了到目前为止支持S.T.Yau的统一化猜想的最强联系之一，该猜想指出，无论地平线上的曲率行为如何，这一结果都是正确的。我们对上述问题的研究是基于对非紧Kähler流形上的Ricci流的深入研究。*我研究的第二个目标涉及确定底层空间何时可以呈现出与具有美丽数学描述的几何对象相对应的某些理想形状的问题，但对于哪些具体实例很难构建。完整的爱因斯坦度规描述了这样一类形状，而一个基本的问题是，给定的完备Kähler流形是否允许爱因斯坦度规。回答这个问题的一个很好的方法是证明当给定的度规沿着利玛窦流动变形时收敛到一个极限，因为这个流动的任何极限都必然是爱因斯坦的。一个密切相关的问题是Kähler Ricci流在爱因斯坦度规下的稳定性。在这里，我们考虑Kähler Ricci流从一种度量开始的收敛，该度量是先验的，接近于爱因斯坦度量。这样的结果对于在更一般的假设下理解收敛是基本的，关键是允许尽可能弱的上述“封闭性”的概念。*一个平行的目标是研究Ricci流如何变形在无界曲率意义上表现不佳的形状。这是对经典的Ricci流理论的扩展，它需要有界曲率，这样的研究是全面解决上述非紧流形上的几何问题的关键。*

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.*** **