Paper · Apr 11, 2016
On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature
As on arXiv
Abstract
We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons which have harmonic Weyl curvature, i.e. . Roughly speaking, we prove that the soliton metric is locally isometric to one of the following four types: an Einstein metric, the product of the Euclidean metric and a 2-d Riemannian manifold of constant curvature , a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works and Derdziński's study on Codazzi tensors. Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with . For shrinking case, it reproves the rigidity result in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally-flat ones with . We also characterize {\it locally} 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.
- Submitted
- Apr 11, 2016
- Subjects
- math.DG
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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
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 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.