We review some constructions and properties of complex manifolds admitting
pluriclosed and balanced metrics. We prove that for a 6-dimensional
solvmanifold endowed with an invariant complex structure J having
holomorphically trivial canonical bundle the pluriclosed flow has a long time
solution for every invariant initial datum. Moreover, we state a new conjecture
about the existence of balanced and SKT metrics on compact complex manifolds.
We show that the conjecture is true for nilmanifolds of dimension 6 and 8 and
for 6-dimensional solvmanifolds with holomorphically trivial canonical bundle.Comment: 16 pages. To appear in a special issue of the Journal of Geometry and
Physic
