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

LAFUENTE, RAMIRO A.; ARROYO, ROMINA M.

Melbourne

Workshop; Australian-German Workshop on Differential Geometry in the Large.; 2019

The {\it pluriclosed flow} is a geometric flow that evolves pluriclosed Hermitian structures (i.e. Hermitian structures for which its $2$-fundamental form satisfies $\partial \bar \partial \omega =0$) 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 Lie groups. More precisely, invariant structures on $2$-step nilmanifolds and almost abelian solvmanifolds. We will analyze the flow and explain how a suitable normalization converges to {\it 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 is a joint work with Ramiro Lafuente (The University of Queensland).