Minimal metrics on 6-dimensional complex nilmanifolds

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.