On the symplectic curvature flow for locally homogeneous manifolds

JORGE LAURET, CYNTHIA WILL

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 analmost-K¨ahler manifold called the symplectic curvature flow, in which the metric, thesymplectic structure and the almost-complex structure are all evolving. We study in thispaper different aspects of the flow on locally homogeneous manifolds, including long-timeexistence, solitons, regularity and convergence. We develop in detail two classes of Liegroups, which are relatively simple from a structural point of view but yet geometricallyrich and exotic: solvable Lie groups with a codimension one abelian normal subgroup anda construction attached to each left symmetric algebra. As an application, we exhibita soliton structure on most of symplectic surfaces which are Lie groups. A family ofancient solutions which develop a finite time singularity was found; neither their Chernscalar nor their scalar curvature are monotone along the flow and they converge in thepointed sense to a (non-K¨ahler) shrinking soliton solution on the same Lie group.