On a Classification of 4-d Gradient Ricci Solitons with HarmonicWeyl Curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonicWeyl 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 R-2 x N. of the Euclidean metric and a 2-d Riemannian manifold of constant curvature. lambda not equal 0, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works (in Trans Am Math Soc 364: 2377-2391, 2012; DukeMath J 162: 10031204, 2013) and Derdzinski's study on Codazzi tensors (in Math Z 172: 273-280, 1980). 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 the shrinking case, it re-proves the rigidity result (Fernandez-Lopez and GarciaRio inMath Z 269: 461- 466, 2011; Munteanu and Sesum in J. Geom Anal 23: 539- 561, 2013) 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 locally 4- d (not necessarily complete) gradient Ricci solitons with harmonic curvature.