Einstein solvmanifolds and nilsolitons

TL;DR

This paper reviews recent advances in classifying solvable Lie groups with Einstein metrics, linking the problem to Ricci solitons on nilpotent Lie groups, and summarizes the current status of this research area.

Contribution

It provides an overview of the latest progress and the current understanding of Einstein solvmanifolds and their relation to nilsolitons, highlighting the classification challenges.

Findings

Progress in classifying Einstein solvmanifolds

Equivalence between Einstein metrics and Ricci solitons on nilpotent groups

Current status of classification efforts

Abstract

The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far of noncompact Einstein homogeneous manifolds. The problem turns to be equivalent to the classification of Ricci soliton left invariant metrics on nilpotent Lie groups.