Filiform nilsolitons of dimension 8

TL;DR

This paper classifies certain 9-dimensional Einstein solvmanifolds with 8-dimensional nilradicals, contributing to the understanding of the Alekseevskii Conjecture in differential geometry.

Contribution

It provides a classification of Einstein solvmanifolds of dimension 9 with 7-step nilpotent Lie algebra nilradicals, advancing the study of homogeneous Einstein spaces.

Findings

Identified all filiform nilsolitons of dimension 8

Classified Einstein solvmanifolds with specified nilradicals

Supported the Alekseevskii Conjecture in this context

Abstract

A Riemannian manifold (M,g) is said to be Einstein if its Ricci tensor satisfies ric(g) = cg, for some real number c. In the homogeneous case, a problem that is still open is the so called Alekseevskii Conjecture. This conjecture says that any homogeneous Einstein space with negative scalar curvature (i.e. c < 0) is a solvmanifold: a simply connected solvable Lie group endowed with a left invariant Riemannian metric. The aim of this paper is to classify Einstein solvmanifolds of dimension 9 whose nilradicals are 7-step nilpotent Lie algebras of dimension 8.