Symplectic curvature flow on Lie groups

WILL, C; LAURET, JORGE

JOURNAL OF SYMPLECTIC GEOMETRY

INT PRESS BOSTON, INC

Año: 2017 vol. 15 p. 1 - 49

1527-5256

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 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-K"ahler) shrinking soliton solution on the same Lie group.