Ricci soliton solvmanifolds

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.

Locations

• arXiv (Cornell University) - View - PDF
• Journal für die reine und angewandte Mathematik (Crelles Journal) - View