Rigidity of four-dimensional gradient shrinking Ricci solitons

Abstract

Abstract Let ( M , g , f ) {{(M,g,f)}} be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric + ∇ 2 ⁡ f = λ ⁢ g {{\mathrm{Ric}+\nabla^{2}f=\lambda g}} , where λ {{\lambda}} is a positive real number. We prove that if M {{M}} has constant scalar curvature S = 2 ⁢ λ {{S=2\lambda}} , it must be a quotient of 𝕊 2 × ℝ 2 {{\mathbb{S}^{2}\times\mathbb{R}^{2}}} . Together with the known results, this implies that a four-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton ℝ 4 {{\mathbb{R}^{4}}} , 𝕊 2 × ℝ 2 {{\mathbb{S}^{2}\times\mathbb{R}^{2}}} or 𝕊 3 × ℝ {{\mathbb{S}^{3}\times\mathbb{R}}} .