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 the 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.

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.

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.

As 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 the 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.

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 the 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.