Geometry and classifications of Ricci solutions

对几何对象进行分类和分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。这是一个经典的结果，即每个二维曲面都带有一定的常曲率度量。对于三维情况，问题变得困难得多。佩雷尔曼在大约十年前成功地解决了这个问题，这是他的一项非常著名的成就。他采用了汉密尔顿开发的一种方法，汉密尔顿设计了一种通过一组微分方程式来变形任意度量的方法。对于佩雷尔曼来说，理解这组方程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和更远的维度。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流的奇性分析中起着核心作用。这个项目的主要主题是研究收缩的利玛窦孤子。在第二个维度和第三个维度，它们已经被完全归类。这种分类在解决三维流形的Poincare猜想和几何化猜想方面有着重要的应用。目前的项目旨在获得四维完全梯度收缩Ricci孤子的分类，建立在包括曲率估计和无穷远结构描述在内的最新进展的基础上。该项目的另一个目标是为高维梯度收缩Ricci孤子的可能分类奠定一些基础。

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.