# Hermitian Calabi functional in complexified orbits

## TL;DR

This paper studies the Hermitian Calabi functional on the space of compatible almost complex structures, providing explicit formulas for its Hessian at extremal metrics, proving semi-positivity at critical points, and analyzing flow properties.

## Contribution

It derives an explicit Hessian formula for the Hermitian Calabi functional and establishes semi-positivity at extremal metrics within complexified orbits, extending K"ahler case results.

## Key findings

- Hessian of Hermitian Calabi functional is semi-positive definite at critical points.
- Explicit formula for the Hessian at extremal almost K"ahler metrics.
- Weak parabolicity of the Hermitian Calabi flow.

## Abstract

Let $(M,\omega)$ be a compact symplectic manifold. We denote by $\ac$ the space of all almost complex structure compatible with $\omega$. $\ac$ 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 K\"ahler 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 K\"ahler case. We also show weak parabolicity of the Hermitian Calabi flow.