On the classification of gradient Ricci solitons

By Peter Petersen, W. Wylie. 2007

225 citations recorded by the source corpus.

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n , S n 1 R and R n . This gives a new proof of the Hamilton‐Ivey‐Perelman classification of 3‐dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of H n , H n 1 R, R n , S n 1 R or S n . 53C25

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/6660e300ba0179c284e5b093a607362f91aed159

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

GRADIENT SHRINKING SOLITONS WITH VANISHING WEYL TENSOR (2008, methodology): “Finally, by [Ni and Wallach 2008] or [Petersen and Wylie 2008], we get the classification theorem, Theorem 1.2.”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “A proof in the compact case can be found in the paper of Eminenti-La Nave-Mantegazza [28] and, in the noncompact case, in the combined work of Ni-Wallach [46] and Zhang [56] (see also the papers of Petersen-Wylie [49] and Cao-Wang-Zhang [18]).”
On Bach-flat gradient shrinking Ricci solitons (2011, background): “…round sphere S n , or the Gaussian shrinking soliton R n , or the round cylinder S n − 1 × R (ﬁrst due to Z. H. Zhang [26] based on the work of Ni-Wallach [21], see also the works of Eminenti-La Nave-Mantegazza [15], Petersen-Wylie [24], X. Cao, B. Wang and Z. Zhang [10], and Munteanu-Sesum [20]).”
Ricci solitons on Lorentzian manifolds with large isometry groups (2010, background): “This leads to new examples of Lorentzian locally conformally flat Ricci solitons without Riemannian analogue (see [17] for the Riemannian case).”
A remark on Einstein warped products (2010, methodology): “Following the terminology introduced in [Petersen and Wylie 2010], the f -Laplacian on the weighted manifold (M, gM , e f dvol) is the diffusion-type operator defined by 1 f u = e f div(e−f∇u).”
On Locally Conformally Flat Gradient Shrinking Ricci Solitons (2008, background): “Such classification has also been considered in [ELM06], [NW07] and [ PW07 ].”
Remarks on non-compact gradient Ricci solitons (2009, methodology): “The authors would like to thank M. Fernández-López for a careful reading of a preliminary version of the paper and for valuable comments that led, in particular, to a substantial improvement in the case p = 1 of Theorem 2.”
Ricci almost solitons (2010, methodology): “This is the same philosophy used, for instance, in [4,22].”
Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “In the shrinking case, classification results for gradient Ricci solitons have been obtained by many authors under various curvature conditions on the Weyl tensor, e.g., [7; 9; 10; 17; 20; 21; 24; 28; 29; 30; 31; 33; 35; 36], etc.”
Geometry of shrinking Ricci solitons (2014, unknown): “For proofs, one may consult [25].”
On Gradient Ricci Solitons (2009, methodology): “2 in [28] for the f−Laplacian case, it follows that |∇u| = C on M .”
On the geometry of gradient einstein-type manifolds (2014, unknown): “[25], [28], [39], [22], [38], [48], [37], [40], [10], [7], [14], [42], [11], [13], [8], [6], [33], [5] and references therein).”