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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 by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kahler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedulli and L. Vezzoni and of the anomaly flow of D. H. Phong, S. Picard and X. Zhang on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kahler metrics are fixed points.
We consider seven-dimensional unimodular Lie algebras $mathfrak{g}$ admitting exact $G_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(mathfrak{g})$. We discuss some examples, both in the case when $b_2(mathfrak{
In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $mathcal{C}$-subsolution. As an applicat
In this paper, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the $C^2$-estimate of canon
We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G$_2$-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G$_2$-structures, up t
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.