Rigidity of four-dimensional gradient shrinking Ricci solitons

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 {Ric+^{2f= g}}, where λ {} is a positive real number. We prove that if M {M} has constant scalar curvature S = 2 ⁢ λ {S=2}, it must be a quotient of 𝕊 2 × ℝ 2 {S^{2^2}}. 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 {R^{4}}, 𝕊 2 × ℝ 2 {S^{2^2}} or 𝕊 3 × ℝ {S^{3}}.