infimum of the Calabi energy in a given Kahler class

Question

Given a compact Kahler manifold $M^n$ and a Kahler class $\Omega$. We have the Calabi energy (or Calabi functional) $$\textrm{Ca}(\omega)=\int_Ms^2(\omega)\omega^n,\qquad \forall \omega\in\Omega.$$ Here $s(\omega)$ is the scalar curvature with respect to the metric $\omega$. My question is whether or not there is an explict expression of $$\textrm{inf}_{\omega\in\Omega}\textrm{Ca}(\omega).$$

Here by "explicit" I mean that if it can be expressed in terms of some well-known or familar quantities in Kahler geometry.

According to my best knowledge, A.D. Hwang and Xiuxiong Chen have a result that $\textrm{Ca}(\omega)$ has a lower bound in terms of the Calabi-Futaki functional evaluted at the extremal vector field and the equality holds iff there exists an extremal metric in $\Omega$. So according to their result, $\textrm{inf}_{\omega\in\Omega}\textrm{Ca}(\omega)$ also has this as its lower bound.

Answer 1

If $\Omega$ is a rational Kähler class, so $M$ is projective algebraic, then Donaldson was the first to prove the Chen-Hwang lower bound that you stated. In fact, in this case he proved more, namely that $$\inf_{\omega\in\Omega}\mathrm{Ca}(\omega)\geq \sup_{\mathfrak{X}}\Psi(\mathfrak{X}),$$ where $\mathfrak{X}$ is any test configuration for $(M,\Omega)$, and $\Psi(\mathfrak{X})$ is a suitable normalization of the Futaki invariant of $\mathfrak{X}$ (see the paper of Donaldson for definitions). The point is that if $(M,\Omega)$ is K-unstable, then the supremum is strictly positive (by definition), and so you conclude that there is no constant scalar curvature Kähler metric in the class $\Omega$.

In the same paper (p.455) Donaldson conjectures that equality above should always hold. This would be the sharp lower bound that you are looking for, and it is still open in general. Very recently, as a consequence of work of Chen, Donaldson, Sun, Tian and Li, it was shown that equality holds when $M$ is Fano, $\Omega$ is the anticanonical class, and $(M,\Omega)$ is K-stable or K-semistable (i.e. the case when the supremum on the RHS is zero).

Therefore there is no general "simple" formula for the lower bound of the Calabi energy, and this is related to the difficult open problem of relating existence of constant scalar curvature Kähler to algebro-geometric stability. See for example these surveys for more on this problem.

By the way, Donaldson's inequality was inspired by a similar identity proved by Atiyah-Bott for holomorphic vector bundles on algebraic curves, with the infimum of the Calabi energy replaced by the infimum of the $L^2$ norm of the curvature of all compatible unitary connections. This identity was recently extended by A.Jacob to all holomorphic vector bundles on compact Kähler manifolds.