I am currently reading the proof of convergence of Kähler-Ricci flow in the case $c_1(M)=0$ from Song, Weinkove. On page 45 he defines the term Mabuchi's $K$-energy functional:
$$ \frac{d}{dt}\mathrm{Mab}_{\omega_0}(\phi_t)=-\int_M\dot{\phi}_tR_{\phi_t}\omega_{\phi_t}^n$$ on the space: $$\mathrm{PSH}(M, \omega_0)=\left\{\phi\in C^\infty(M)\mid \omega_0+\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\phi>0\right\}$$ and where $\omega_{\phi_t}=\omega_0+\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\phi_t$ and $R_{\phi_t}$ is the scalar curvature of $\omega_{\phi_t}$.
Then he directly claims that if $\phi_\infty$ is a critical point of $\mathrm{Mab}_{\omega_0}$ then $\omega_\infty$ has zero scalar curvature. I do not see why is that so. I am not an expert in this realm, so I might miss some well-known facts or obvious results.
Any comment is welcome!