The long-time behaviour of the pluriclosed flow on almost abelian Lie groups.

The pluriclosed flow is a geometric flow that evolves pluriclosed Hermitian structures (i.e. Hermitian structures for which its 2-fundamental form satisfies ∂∂ω¯ = 0) in a given complex manifold. The aim of this talk is to discuss the asymptotic behaviour of the pluriclosed flow in the case of left-invariant structures on almost abelian Lie groups (i.e. Lie groups whose Lie algebra has an abelian ideal of codimension one). We will analyze the flow and explain how a suitable normalization converges to pluriclosed solitons, which are self-similar solutions to the flow. Moreover, we will show that some of those limits are shrinking solitons, which is an unexpected feature in the solvable case. We will also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.

This talk is based on joint work with Ramiro Lafuente (The University of Queensland).