Rigidity of complete gradient steady Ricci solitons with harmonic Weyl curvature

Fengjiang Li

Vol. 335 (2025), No. 2, 323–353

DOI: 10.2140/pjm.2025.335.323

Abstract

We investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. More precisely, we prove that an $n$-dimensional ($n\ge 5$) complete noncompact gradient steady Ricci soliton with harmonic Weyl tensor and multiply warped product metric is either Ricci flat or isometric to the Bryant soliton up to scaling. 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 and multiply warped product metric.

Keywords

gradient Ricci solitons, harmonic Weyl curvature, Codazzi tensor

Mathematical Subject Classification

Primary: 53C20

Secondary: 53C25, 53E20

Milestones

Received: 27 February 2025

Revised: 19 March 2025

Accepted: 8 April 2025

Published: 30 April 2025