Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (5): 1011-1017.

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

Zhuhong Zhang()   

  1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631
  • Received:2018-08-30 Online:2019-10-26 Published:2019-11-08


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 $\overset{\circ }{\mathop{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

CLC Number: 

  • O186.12