Hermitian Calabi functional in complexified orbits

Abstract

Let $(M,ω)$ be a compact symplectic manifold. We denote by $\ac$ the space of all almost complex structure compatible with $ω$. $\ac$ has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit formula of the Hessian of Hermitian Calabi functional at an extremal almost Kähler metric in $\ac$. We prove that the Hessian of Hermitian Calabi functional is semi-positive definite at critical point when restricted to a complexified orbit, as corollaries we obtain some results analogy to Kähler case. We also show weak parabolicity of the Hermitian Calabi flow.