Paper · Jul 23, 2023
Classification of Gradient Ricci solitons with harmonic Weyl curvature
As on arXiv
Abstract
We make classifications of gradient Ricci solitons with harmonic Weyl curvature. As a local classification, we prove that the soliton metric is locally isometric to one of the following four types: an Einstein manifold, the Riemannian product of a Ricci flat manifold and an Einstein manifold, a warped product of and an Einstein manifold, and a singular warped product of and a Ricci flat manifold. Compared with the previous four-dimensional study in, we have developed a novel method of {\it refined adapted frame fields} and overcome the main difficulty arising from a large number of Riemmannian connection components in dimension. Next we have obtained a classification of {\it complete} gradient Ricci solitons with harmonic Weyl curvature. For the proof, using the real analytic nature of and , we elaborate geometric arguments to fit together local regions.
- Submitted
- Jul 23, 2023
- Subjects
- math.DG
Comments: 41 pages
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Read around it
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
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
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.