Ricci soliton solvmanifolds

JORGE LAURET

Oberwolfach, Alemania

Congreso; Geometrie (Oberwolfach); 2008

Mathematisches Forschungsinstitut Oberwolfach

All known examples of nontrivial homogeneous Ricci solitons are left-invariantmetrics on simply connected solvable Lie groups whose Ricci operator is a multipleof the identity modulo derivations (called {\it solsolitons}, and {\it nilsolitons}in the nilpotent case).  The tools from geometric invariant theory used to studyEinstein 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 simpleconstruction from a nilsoliton $N$ together with any abelian Lie algebra ofsymmetric derivations of its metric Lie algebra $(\ngo,\ip)$. The followinguniqueness result is also obtained: a given solvable Lie group can admit at most onesolsoliton up to isometry and scaling.  As an application, solsolitons of dimension$\leq 4$ are classified.