Hermitian Calabi functional in complexified orbits

Abstract

Let [Formula: see text] be a compact symplectic manifold. We denote by [Formula: see text] the space of all almost complex structure compatible with [Formula: see text]. [Formula: see text] 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 [Formula: see text]. 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.