Gradient Steady Ricci Solitons with Harmonic Weyl Curvature

Fengjiang Li

Published 2021-01-29Version 1

Our main aim in this paper is to investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. More precisely, we prove that an $n$-dimensional ($n\geq 5$) complete noncompact gradient steady Ricci soliton with harmonic Weyl tensor is either Ricci flat or isometric to the Bryant soliton up to scaling. We also derive a classification result for complete noncompact gradient expanding Ricci solitons with harmonic Weyl tensor. Meanwhile, for $n\ge 5$, we provide a local structure theorem for $n$-dimensional connected (not necessarily complete) gradient Ricci solitons with harmonic Weyl curvature, thus extending the work of Kim [31] for $n=4$. Furthermore, a similar method can be applied to treat vacuum static spaces and CPE metrics with harmonic curvature [32, 11], as well as quasi-Einstein manifolds with harmonic Weyl curvature [12].