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 $\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