Four-dimensional complete gradient shrinking Ricci solitons

Abstract

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton ${\mathbb{R}}^4,$ or ${\mathbb{S}}^3\times \mathbb{R}$, or ${\mathbb{S}}^2\times {\mathbb{R}}^2.$ In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

Final Version. To appear in Crelle's Journal (Journal für die reine und angewandte Mathematik)

References in corpus (3)

• Geometry of shrinking Ricci solitons
• The Weyl Tensor of Gradient Ricci Solitons
• A gap theorem of four-dimensional gradient shrinking solitons