Journal of Symplectic Geometry

Volume 15 (2017)

Number 1

On the symplectic curvature flow for locally homogeneous manifolds

Pages: 1 – 49



Jorge Lauret (Universidad Nacional de Córdoba, FaMAF and CIEM, Córdoba, Argentina)

Cynthia Will (Universidad Nacional de Córdoba, FaMAF and CIEM, Córdoba, Argentina)


Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kähler 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 large 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-Kähler) shrinking soliton solution on the same Lie group.

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