Minimal metrics on nilmanifolds

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.