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 ℝ 4 R^{4}, or 𝕊 3 × ℝ S^{3}, or 𝕊 2 × ℝ 2. S^{2^2. } 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.