Minimal metrics on 6-dimensional complex nilmanifolds

TL;DR

This paper classifies all complex structures on 6-dimensional nilpotent Lie groups that admit a minimal compatible metric, which minimizes the Ricci tensor norm among metrics with the same scalar curvature.

Contribution

It provides a complete classification of complex structures on 6D nilpotent Lie groups that support minimal compatible metrics.

Findings

Identified all complex structures with minimal metrics on 6D nilpotent Lie groups

Characterized the geometric properties of these minimal metrics

Established criteria for the existence of minimal metrics in this setting

Abstract

Let (N,J) be a real 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. A left-invariant Riemannian metric on N compatible with J is said to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics on (N,J) with the same scalar curvature. In this paper, we determine all complex structures that admit a minimal compatible metric on 6-dimensional nilpotent Lie groups.