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arXiv · 1405.6065

On the symplectic curvature flow for locally homogeneous manifolds

Jorge Lauret·Cynthia Will
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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.

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Jorge Lauret, Cynthia Will. 2014-05-23. On the symplectic curvature flow for locally homogeneous manifolds. https://arxiv.org/abs/1405.6065

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