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

Abstract

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 ${\lambda } \ne 0$, 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; Duke Math J 162:1003–1204, 2013) and Derdziński’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 (Fernández-López and García-Río in Math 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.