Title: On the classification of four-dimensional gradient Ricci solitons,
Abstract: In this paper, we prove some classification results for four-dimensional
gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci
soliton with $div^4Rm^\pm=0$, 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 $div^4W^\pm=0$. 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.