On the classification of four-dimensional gradient Ricci solitons

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $di{v}^{4}R{m}^{\pm }=0$, we show that it is either Einstein or a finite quotient of ${\mathbb{R}}^4$, ${\mathbb{S}}^2\times {\mathbb{R}}^2$ or ${\mathbb{S}}^3\times \mathbb{R}$. The same result can be obtained under the condition of $di{v}^4{W}^{\pm }=0$. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.