数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1011-1017.

• 论文 • 上一篇    

半正迷向曲率的四维Shrinking Gradient Ricci Solitons

张珠洪   

  1. 华南师范大学数学科学学院 广州 510631
  • 收稿日期:2018-08-30 修回日期:2019-01-04 发布日期:2019-11-08
  • 通讯作者: 张珠洪 E-mail:juhoncheung@sina.com

Four-Dimensional Shrinking Gradient Ricci Solitons with Half Positive Isotropy Curvature

Zhang Zhuhong   

  1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631
  • Received:2018-08-30 Revised:2019-01-04 Published:2019-11-08

摘要: 该文主要研究一类四维shrinking gradient Ricci solitons,它们具有半正迷向曲率(half-PIC).该文证明了traceless Ricci曲率Ric的界可以控制Weyl张量的自对偶部分W+或反自对偶部分W-的界.特别的,该文可以给出下述命题一个新的简单的证明:任何一个具有half-PIC的可定向四维Einstein流形,是半共形平坦的,从而一定等距于S4CP2.作者还证明了在shrinking gradient Ricci soliton上成立一个更一般的结论.

关键词: Gradient Ricci solitons, Einstein流形, 半正迷向曲率, 极值原理

Abstract: In this paper, we will study four-dimensional shrinking gradient Ricci solitons with half positive isotropy curvature (half-PIC). We will show that, the bound of the traceless Ricci curvature Ric will control the bound of the self-dual part of the Weyl tensor W+ or the antiself-dual part W-. In particular, we will give a new and simpler proof of the following theorem:Any oriented four-dimensional Einstein manifold with half-PIC must be half conformally flat, and therefore isometric to S4 or CP2 with standard metric. A more general result on shrinking gradient Ricci solitons was gave.

Key words: Gradient Ricci soliton, Einstein manifold, Half positive isotropy curvature, Maximum principle

中图分类号: 

  • O186.12