Ricci soliton solvmanifolds

Details Ricci soliton solvmanifolds Ricci soliton solvmanifolds

2010-02-02

arxiv.org/abs/1002.0384v1

Mathematics

Differential Geometry

18 pages, to appear in Crelle's Journal

Scientific paper

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 together with any abelian Lie algebra of symmetric derivations of its metric Lie algebra. 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 at most 4 are classified.

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