Hessians of the Calabi functional and the norm function (Q2502302)

Let \(X\) be a Kähler manifold of dimension \(n\) with a fixed Kähler class \([\omega_0]\). One can identify it with \[ K= \{\varphi: \omega_0+ i\partial\overline\partial\varphi> 0\}/\mathbb{R}. \] The Calabi functional on \(K\) is defined for \(\omega= \omega_0+ i\partial\overline\partial\varphi\) as \[ C(\omega)= \int_X s^2{\omega^n\over n!}, \] where \(s\) is the scalar curvature of the metric \(\omega\). The author first gives a proof of the result of Calabi which says that the Hessian of \(C\) is nonnegative at critical points, i.e., at extremal metrics. The main result of the paper is an extension of Calabi's result to a general context, i.e., for any reductive group action on a non-Kähler manifold admitting a moment map. He shows that the Hessian of the norm function is always nonnegative along a complex orbit at a critical point.