A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Let $(M^4,g,f)$ be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+{\mathrm{\nabla }}^{2}f=\frac{1}{2}g$. If its scalar curvature is $1$, Cheng-Zhou \cite{Cheng-Zhou} proved that it is a finite quotient of ${\mathbb{R}}^2\times {\mathbb{S}}^2$. In this note we present an alternative proof by analyzing the asymptotic geometry at infinity.