Ricci soliton solvmanifolds

By Jorge Lauret. 2010

153 citations recorded by the source corpus.

Abstract All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the nilpotent case). The tools from geometric invariant theory used to study Einstein solvmanifolds, turned out to be useful in the study of solsolitons as well. We prove that, up to isometry, any solsoliton can be obtained via a very simple construction from a nilsoliton N together with any abelian Lie algebra of symmetric derivations of its metric Lie algebra (𝔫, 〈·, ·〉). The following uniqueness result is also obtained: a given solvable Lie group can admit at most one solsoliton up to isometry and scaling. As an application, solsolitons of dimension ≦ 4 are classified.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b362d30f99393005b91bae85117667b1b79ceb78

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In Applied Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

On the Chern‐Ricci flow and its solitons for Lie groups (2013, result): “Recall that this is in clear contrast to the behavior of other curvature flows on solvmanifolds like the Ricci flow (see [12]) and the symplectic curvature flow (see [15]).”
The Ricci flow for simply connected nilmanifolds (2010, background): “It is proved in [24] that all solvsolitons can be obtained”
Laplacian Flow of Closed G$$_2$$2-Structures Inducing Nilsolitons (2013, background): “Therefore, as it was noticed by Lauret in [22] we have the following”
Structure of homogeneous Ricci solitons and the Alekseevskii conjecture (2012, result): “As a first step toward a possible resolution of the above conjecture, we obtain in the present paper some structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize the results on Einstein solvmanifolds obtained in [13, 23], as well as those on solvsolito”