Paper · Jan 29, 2021
Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature
As on arXiv
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 -dimensional () 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 , we provide a local structure theorem for -dimensional connected (not necessarily complete) gradient Ricci solitons with harmonic Weyl curvature and multiply warped product metric.
- Submitted
- Jan 29, 2021
- Last revised
- May 17, 2023 (v3)
- Subjects
- math.DG
Comments: 27 pages. arXiv admin note: text overlap with arXiv:1604.02827 by other authors
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
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Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12B.1κ-Solutions
Ancient, nonnegatively curved, noncollapsed: the models for singularities.