Four-dimensional complete gradient shrinking Ricci solitons
Huai-Dong Cao, Ernani Ribeiro, Detang Zhou
Abstract
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>4</m:mn> </m:msup> </m:math> {\mathbb{R}^{4}} , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>𝕊</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>×</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> {\mathbb{S}^{3}\times\mathbb{R}} , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>𝕊</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> {\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.