Ricci孤立子的曲率及势函数

Curvature and Potential Function of Ricci Solitons

摘要: 过去几十年中，人们越来越关注满足特殊结构方程度量的黎曼流形的研究。其中一个最重要的例子是Ricci流和Ricci孤立子。Ricci流是研究黎曼流形最有力的工具之一，它在Hamilton和Perelman证明Poincaré猜想过程中起着关键作用，并且广泛用于研究流形的拓扑结构、几何性质和其它复杂结构。Ricci流方程本身作为偏微分方程的研究也十分重要，它给出了关键度量的规范方法。关于Ricci孤立子有两个重要的研究方向，一是研究黎曼流形的Ricci孤立子结构对拓扑结构的影响，另一个是研究它在几何学中的影响。本文，我们将归纳总结Ricci孤立子曲率及势函数的估计结果。

Abstract: In the last decades, there has been an increasing interest in the study of Riemannian manifolds endowed with metrics satisfying special structural equation. One of the most important examples is represented by Ricci flow and Ricci solitons that has become the subject of rapidly increasing investigation since the appearance of the seminal works. It plays a key role in Hamilton and Pe-relman’s proof of the Poincaré conjecture, and has been widely used to study the topology, geometry and complex structure of manifolds. The Ricci flow equation is of own interest as a geometric partial differential equation. It gives a canonical way of a critical metric. There are two important aspects of Ricci solitons. One looks at the influence on the topology by the Ricci soliton structure of the Riemannian manifold, and the other looks at its influence in its geometry. In this paper, we are interested in summarizing some new results about the curvature and potential function estimates of Ricci solitons.