Ricci孤立子的刚性及体积增长
Rigidity and Volume Growth of Ricci Solitons
DOI: 10.12677/PM.2019.91002, PDF,    科研立项经费支持
作者: 李金楠, 高 翔*：中国海洋大学，数学科学学院，山东 青岛
关键词: Ricci孤立子；刚性；体积增长；Ricci Soliton； Rigidity； Volume Growth
摘要: 20世纪80年代Hamilton提出Ricci流的概念并用于解决Poincaré猜想后，Ricci流的自相似解(即Ricci孤立子)的分类及几何结构的研究得到迅速发展。梯度Ricci孤立子为刚性的若它等距于N×Rk的一个有限商空间，其中N 为爱因斯坦流形。测地球的体积增长是研究流形及Ricci孤立子重要的几何性质，体积增长率也是重要的几何不变量。本文将系统阐述Ricci孤立子的基本发展、Ricci孤立子的刚性及体积增长结果，给出完备非紧致梯度Ricci孤立子线性或欧氏体积增长的结论。
Abstract: After the concept of Ricci flow was proposed by Hamilton in the 1980s and applied to solve the Poincaré conjecture, the classification and geometry of the self-similar solution of Ricci flow (i.e., Ricci soliton) developed rapidly. A gradient soliton is said to be rigid if it is isometric to a quotient of N×Rk, where N is an Einstein manifold. Volume growth of geodesic balls is an important geometric property for studying manifolds and Ricci soliton and the volume growth rate is an im-portant geometric invariant of manifolds. In this paper, we will systematically summarize the basic development of Ricci solitons, the rigidity and volume growth results of Ricci solitons, and give the conclusion that the complete non-compact gradient Ricci solitons have linear or Euclidean volume growth.×
文章引用：李金楠, 高翔. Ricci孤立子的刚性及体积增长[J]. 理论数学, 2019, 9(1): 11-19. https://doi.org/10.12677/PM.2019.91002

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