Evolution Equations and Breathers for Symplectic Curvature Flow

Abstract

We begin by demonstrating how Blair's identity [5] and a generalization are central to symplectic curvature flow. This leads to similar identities for closed almost Kahler manifolds involving quantities relevant to this flow. We use these ideas to give basic characterizations depending on the nature of the first Chern class for the non-existence of certain kinds of breathers and solitons for symplectic curvature flow. In particular, we study the case of symplectically minimal, closed almost Kahler 4-manifolds and give a basic characterization of existence time, the nature of singularities, and the types of admissible breathers and solitons depending on the symplectic Kodaira dimension and a type of averaged scalar curvature. In particular, we obtain a monotonicity result for this type of averaged scalar curvature for closed, symplectically minimal almost Kahler manifolds with symplectic Kodaira dimension 2. Finally, we calculate the evolution equation by symplectic curvature flow of a functional analogous to Perelman's F-functional. We do this in multiple different ways and relate an expression involving the contorsion tensor of the Chern connection to the Levi-Civita derivative of the almost complex structure.