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Are there any results concerning the existence of extremal Kähler metrics on Calabi-Yau manifolds (ie manifolds that also admit a Ricci-flat Kähler metric)? From what I understand the sign of the scalar curvature of an extremal Kähler metric is determined by the first Chern class and the Kähler class, so although $c_{1}(M)=0$ for a manifold admitting a Ricci-flat Kähler metric, because of the dependence on the Kähler class it seems possible that such a manifold could also admit an extremal Kähler metric with nonzero scalar curvature.

The specific case I am envisioning is the manifold being a hypersurface in $\mathbb{CP}^{n}$ with $c_{1}(M)=0$ but I would also be interested in Calabi-Yau manifolds produced by other contexts (blowing up etc)

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By Chen and Tian, if $X$ is a compact Kahler manifold and $\omega_1$ and $\omega_2$ are two extremal Kahler metrics such that $[\omega_1] = [\omega_2] \in H^2(X,\mathbb R)$, then there exists a holomorphic automorphism of $X$ such that $f^*\omega_1 = \omega_2$.

Thus if $\omega$ is an extremal Kahler metric on a compact Ricci-flat Kahler manifold $X$, there exists a unique Ricci-flat Kahler metric $\omega'$ in the Kahler class $[\omega]$, and a holomorphic automorphism $f$ such that $f^*\omega = \omega'$. But then $\omega$ itself is Ricci-flat, and in fact $\omega = \omega'$ by uniqueness.

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