Ricci soliton solvmanifolds

TL;DR

This paper classifies and constructs Ricci soliton solvmanifolds, showing that all such structures can be derived from nilsolitons and abelian derivations, with uniqueness results for their existence.

Contribution

It provides a simple construction method for solsolitons from nilsolitons and proves a uniqueness theorem for solsolitons on a given solvable Lie group.

Findings

All solsolitons can be obtained from nilsolitons and abelian derivations.

A solvable Lie group admits at most one solsoliton up to isometry and scaling.

Classification of solsolitons in dimensions up to 4.

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 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.