  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.
