On Gradient Ricci Solitons with Symmetry

By Peter Petersen, W. Wylie. 2007

171 citations recorded by the source corpus.

We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus, the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper "Rigidity of gradient Ricci solitons" to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8176603415acf07bb3943e8de4e0443c963b090a

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Citations in the corpus

Laplacian Flow of Closed G$$_2$$2-Structures Inducing Nilsolitons (2013, background): “2] and in [27], (M, g) is isometric to a quotient of P × Rk , where P is some homogeneous Einstein manifold with positive scalar curvature.”
Ricci solitons on Lorentzian manifolds with large isometry groups (2010, background): “Although Ricci solitons exist on many Lie groups and homogeneous spaces, all homogeneous gradient Ricci solitons are rigid [16] in the Riemannian setting.”
The Ricci flow for simply connected nilmanifolds (2010, background): “1, (ii)) and it can never be gradient (see [23]).”
Gradient Einstein solitons (2012, unknown): “A possible direction of investigation is to prove a rigidity results for noncompact gradient shrinking ρ–Einstein solitons, in analogy with the case of rectifiable gradient Ricci solitons, studied in [30].”
On the classification of gradient Ricci solitons (2007, background): “Conversely, we see that f is recti…able (see [ 30 ]) as”
Rigidity of shrinking Ricci solitons (2011, background): “Rigidity of gradient Ricci solitons has been studied in [13,16,17].”
Homogeneous Ricci solitons (2011, methodology): “instein. Theorem 1.4. Compact homogeneous Ricci solitons are necessarily Einstein. HOMOGENEOUS RICCI SOLITONS 3 Theorem 1.4 is well-known to experts and can be deduced from the work of Petersen-Wylie [PW09] where homogeneous gradient Ricci solitons are studied. To use that work, one must apply a res”
Rigidity of gradient Einstein shrinkers (2013, methodology): “Following Petersen-Wylie [12], we are going to estimate the two terms of the Hessian separately.”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, background): “In addition to the paper of Naber mentioned above, also see [36, 37, 38] for applications of Corollary 5.”
Immortal homogeneous Ricci flows (2017, background): “For expanding homogeneous Ricci solitons there exists already a well-developed structure theory: see [Heb98], [Lau10], [Nik11], [Jab14], [LL14a], [LL14b], [Jab15], [HPW15], [GJ17], [JP16], [AL17]. By [PW09], these solitons are either the product of a negative Einstein metric and a ﬂat Euclidean fact”
Ricci soliton solvmanifolds (2010, background): “ scalar curvature sc(g(t)) satisﬁes that Ric(g) = 0, and consequently g must be ﬂat (see [AK]). In the shrinking case, it follows from [N, Theorem 1.2] that g is of gradient type, and it is proved in [PW] that any homogeneous gradient Ricci soliton is isometric to a quotient of N × Rk, where N is so”
RIGIDITY OF GRADIENT RICCI SOLITONS (2007, methodology): “We refer the reader to [Petersen and Wylie 2007a] for the discussion of this result.”
Homogeneous Ricci Solitons are Algebraic (2013, background): “The theorem above resolves questions raised by Lafuente-Lauret [12] and He-Petersen-Wylie [4].”
Generic Properties of Homogeneous Ricci Solitons (2007, background): “ollowing theorem. Theorem 3.7. Any metric Lie algebra (g;h;i) with trivial Euclidean de Rham factor cannot be a gradient Ricci soliton. Remark 3.8. It follows from a rigidity result of Petersen-Wylie [29] that any homogeneous gradient Ricci soliton must have a non-trivial Euclidean deRham factor. Th”