TITLE: Is there an extremal metric on toric Fano manifolds which have nonzero Futaki invariant?
QUESTION [5 upvotes]: According the work by Wang & Zhu, on toric Fano manifolds there exist Kaehler-Ricci solitons. If Futaki=0, there also exist CSCK metrics. But if the Futaki invariant does not vanish, what about extremal metrics? Is there some counterexample?

REPLY [2 votes]: Some additional comment: 
Let $(X,J,\omega)$ be Fano Kahler manifold with $\omega_0\in [\omega_0]=\kappa$, then by $\partial\bar\partial$-lemma any other Kahler 2-form can be written as $\omega_0+\sqrt{-1}\partial\bar\partial \varphi$. Now the set of Kahler metrics in $[\omega_0]$ can be identified with $$\mathcal H=\{\varphi \in C^\infty(X,\mathbb R)|\omega_0+\sqrt{-1}\partial\bar\partial \varphi>0\}/\mathbb R$$. Note that $T_\varphi\mathcal H=C^\infty(X,\mathbb R)/\mathbb R$
Now, consider the Calabi- functional(which is sort of special case of Yang-Mills functional)
If we denote $S(\omega_\varphi)$ denotes to be the scalar curvature of the metric $\omega_\varphi$ and $\underline S$ be the average scalar
curvature
$$\mathcal C a:\mathcal H\to \mathbb R$$
$$\varphi\to \mathcal Ca(\varphi)=\int_X (S(\omega_\varphi)-\underline S)^2\frac{\omega_\varphi^n}{n!}$$
Now if you take $\psi\in T_\varphi\mathcal H$ , then we can write the variation of Calabi functional via using Lichnerowicz operator $\mathcal D_\varphi=\overline \partial\nabla$ as follows
$$(d\mathcal Ca(\varphi))\psi=2\int_X (\mathcal D_\varphi^*\mathcal D_\varphi S(\omega_\varphi)).\psi\frac{\omega_\varphi^n}{n!}$$ where $\mathcal D_\varphi^*$ is the $L^2(X,\omega_\varphi)$-adjoint of $\mathcal D_\varphi$. As an exercise integrating by part we have $\ker \mathcal D_\varphi^*\mathcal D_\varphi =\ker \mathcal D_\varphi$ 
This means that $\omega_\varphi$ is extremal metric (which means the critical points of Calabi functional) if the gradient of scalar curvature $S(\omega_\varphi)$ define a vector field .
About your question this recent paper https://arxiv.org/pdf/1610.06865.pdf