On the symplectic curvature flow for locally homogeneous manifolds

TL;DR

This paper investigates the symplectic curvature flow on locally homogeneous manifolds, analyzing long-term behavior, solitons, and singularities, with detailed studies on specific Lie groups and applications to symplectic surfaces.

Contribution

It provides new insights into the flow's behavior on certain Lie groups, including existence, solitons, and singularity formation, expanding understanding of symplectic curvature flow in homogeneous settings.

Findings

Existence and convergence results for the flow on specific Lie groups.

Construction of soliton solutions on symplectic Lie groups.

Identification of ancient solutions with finite-time singularities.

Abstract

Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-K\"ahler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, including long-time existence, solitons, regularity and convergence. We develop in detail two classes of Lie groups, which are relatively simple from a structural point of view but yet geometrically rich and exotic: solvable Lie groups with a codimension one abelian normal subgroup and a construction attached to each left symmetric algebra. As an application, we exhibit a soliton structure on most of symplectic surfaces which are Lie groups. A family of ancient solutions which develop a finite time singularity was found; neither their Chern scalar nor their scalar curvature are monotone along the flow and they converge in the pointed sense to a (non-K\"ahler) shrinking soliton solution on the same Lie group.