Rigidity of four-dimensional Gradient shrinking Ricci solitons

Abstract

Let $(M,g,f)$ be a $4$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+{\nabla }^{2}f=\lambda g$, where $\lambda$ is a positive real number. We prove that if $M$ has constant scalar curvature $S=2\lambda$, it must be a quotient of ${\mathbb{S}}^2\times {\mathbb{R}}^2$. Together with the known results, this implies that a $4$-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 ${\mathbb{R}}^4$, ${\mathbb{S}}^2\times {\mathbb{R}}^2$ or ${\mathbb{S}}^3\times \mathbb{R}$.

22 pages. Comments are welcome