On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons $(M, g, f)$ which have harmonic Weyl curvature, i.e. $\delta W=0$. Roughly speaking, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein metric, the product $\mathbb{R}^2 \times N_{\lambda}$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature ${\lambda} \neq 0$, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works \cite{CC1, CC2} and Derdzi\'{n}ski's study on Codazzi tensors \cite{De}. Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with $\delta W=0$. For shrinking case, it reproves the rigidity result \cite{FG, MS} in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally-flat ones with $\delta W=0$. We also characterize {\it locally} 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.