Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Abstract

Let $(M,g,f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+{\nabla }^{2}f=\hat{I}»g$, where $\text{Ric}$ is the Ricci tensor and ${\nabla }^{2}f$ is the Hessian of the potential function $f$. We prove that it is a finite quotient of ${\mathbb{R}}^2\times {\mathbb{S}}^3$ if $M$ has constant scalar curvature $R=3\hat{I}»$.