Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature

TL;DR

This paper proves that complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor and multiply warped product metrics are either Ricci flat or isometric to Bryant solitons, revealing their rigidity.

Contribution

It establishes a rigidity result for such Ricci solitons and provides a local structure theorem for connected cases with harmonic Weyl curvature.

Findings

Complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor are either Ricci flat or Bryant solitons.

A local structure theorem for connected Ricci solitons with harmonic Weyl curvature and multiply warped product metrics.

Results hold for dimensions n ≥ 5.

Abstract

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 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.