Minimal metrics on nilmanifolds

TL;DR

This paper investigates minimal metrics on nilmanifolds, showing their uniqueness, existence, and relation to Einstein solvmanifolds, and provides explicit examples and classifications of such metrics.

Contribution

It offers new explicit examples of minimal metrics on nilpotent Lie groups and explores their connection to Einstein solvmanifolds and special geometric structures.

Findings

Minimal metrics are unique up to isometry and scaling.

Several explicit examples of minimal metrics are constructed.

The paper lists all known Einstein solvmanifolds.

Abstract

A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If $N$ is endowed with an invariant symplectic, complex or hypercomplex structure, then minimal compatible metrics are also unique up to isometry and scaling. The aim of this paper is to give more evidence of the existence of minimal metrics, by presenting several explicit examples. This also provides many continuous families of symplectic, complex and hypercomplex nilpotent Lie groups. A list of all known examples of Einstein solvmanifolds is also given.