Journal of Symplectic Geometry

Volume 15 (2017)

Number 1

On the symplectic curvature flow for locally homogeneous manifolds

Pages: 1 – 49

DOI: http://dx.doi.org/10.4310/JSG.2017.v15.n1.a1

Authors

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

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

Abstract

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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Published 28 April 2017