Geometry and classifications of Ricci solutions

对几何对象进行分类和归类是数学中的一个中心主题。这样做的一种有效方法是为每个对象规定一个规范度量，即测量对象中距离的规范方法。每个二维曲面都带有一个具有常曲率的度量，这是一个经典的结果。对于三维情况，问题变得困难得多。这是一个非常著名的成就佩雷尔曼谁成功地解决了这个问题约十年前。他采用了一种方法开发的汉密尔顿谁设计了一种方式变形任意度量通过一组微分方程。理解佩雷尔曼方程组的关键一步是对所谓的自相似解进行分类。目前的项目旨在将佩雷尔曼的工作扩展到四维及更远的空间。预计这些结果将有助于四维空间的研究。 Ricci孤子作为Ricci流的自相似解，在Ricci流的奇异性分析中起着重要的作用。本项目的主要内容是研究收缩的Ricci孤子。在二维和三维空间中，它们被完全分类。这种分类在Poincare猜想的求解和三维流形的几何化中有重要的应用。目前的项目旨在获得一个四维完全梯度收缩Ricci孤子的分类，建立在最近的进展，包括曲率估计和无穷远结构的描述。本项目的另一个目标是为高维梯度收缩Ricci孤子的分类奠定基础。

It is a central theme in mathematics to categorize and classify geometric objects. One effective way of doing so is to prescribe a canonical metric for each object, namely a canonical way to measure the distance in an object. It is a classical result that each two dimensional surface carries a certain metric with constant curvature. For the three dimensional case, the problem becomes vastly more difficult. It was a much celebrated achievement of Perelman who successfully tackled the problem about ten years ago. He adopted an approach developed by Hamilton who has devised a way of deforming an arbitrary metric via a set of differential equations. A crucial step in understanding this set of equations for Perelman is to classify the so-called self-similar solutions. The current project aims to extend Perelman's work to dimension four and beyond. It is expected that the results will aid in the study of four dimensional spaces. Ricci solitons, as self-similar solutions to the Ricci flows, play a central role in the singularity analysis of the Ricci flows. The main theme of this project is to study shrinking Ricci solitons. In dimensions two and three, they have been completely classified. The classification has found important applications in the resolution of the Poincare and geometrization conjecture for three dimensional manifolds. The current project aims to obtain a classification for four dimensional complete gradient shrinking Ricci solitons, building upon the recent progress including curvature estimates and a description of the structure at infinity. Another goal of the project is to lay some foundation toward a possible classification for high dimensional gradient shrinking Ricci solitons.