Almost Abelian Lie algebras admitting Astheno-Kahler and Balanced structures

Abstract

This thesis systematically studies almost Abelian Lie algebras admitting astheno-Kähler and balanced Hermitian structures. An almost Abelian Lie algebra is defined by the existence of a codimension-one Abelian ideal, and such algebras offer a tractable yet rich framework for exploring complex geometric structures. Focusing on real eight-dimensional almost Abelian Lie algebras, the work develops a comprehensive framework for classifying and analysing left-invariant Hermitian structures. The initial chapters provide the necessary background in complex geometry and Lie theory. Key concepts such as complex manifolds, Hermitian metrics, and the integrability of almost complex structures via the Newlander-Nirenberg theorem are reviewed. Building on these foundations, the thesis introduces Hermitian structures on almost Abelian Lie algebras and derives explicit algebraic criteria for balanced and astheno-Kähler metrics. In particular, balanced metrics are characterised by vanishing the Lee form. At the same time, the astheno-Kähler condition is defined via the vanishing of the $\partial\bar{\partial}$ operator acting on a suitable power of the fundamental form. Overall, the results clearly classify eight-dimensional almost Abelian Lie algebras that support these special Hermitian structures. The study further explores geometric flows on these Lie algebras. Using the bracket flow technique, the evolution equations of left-invariant metrics are reformulated in an algebraic setting. A detailed analysis of the balanced flow is carried out, demonstrating how it preserves the balanced condition and aids in the approach to canonical metric structures. The interplay between the algebraic data and the dynamic behaviour of the flow yields criteria for long-time existence and stability.