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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. $δW=0$. 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_λ$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature $λ \neq 0$, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works \cite{CC1, CC2} and Derdziński'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 $δW=0$. 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 $δW=0$. We also characterize {\it locally} 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.