The long-time behavior of the homogeneous pluriclosed flow

Abstract

We study the asymptotic behavior of the pluriclosed flow in the case of left-invariant Hermitian structures on Lie groups. We prove that solutions on 2-step nilpotent Lie groups and on almost-abelian Lie groups converge, after a suitable normalization, to self-similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We 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.

26 pages, 1 table

References in corpus (4)

• Special Hermitian metrics on compact solvmanifolds
• Immortal homogeneous Ricci flows
• Geometric flows and their solitons on homogeneous spaces
• A remark on the Bismut-Ricci form on 2-step nilmanifolds

Cited by in corpus (7)

• Generalized Kähler almost abelian Lie groups
• Hermitian Curvature flow on unimodular Lie groups and static invariant metrics
• The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds
• Expanding solitons to the Hermitian curvature flow on complex Lie groups
• Hermitian curvature flow on complex locally homogeneous surfaces
• Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups
• A note on $p$-Kähler structures on compact quotients of Lie groups