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We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify 6-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [26] that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with leftinvariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [6] and of the anomaly flow [47] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.

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We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of [Gar19]. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group.

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.