Calabi's conjecture on prescribed Ricci forms

Let $M$ be a compact Kähler manifold with Kähler metric $\omega$, and let $\alpha$ be a real $(1,1)$-form representing $c_{1}(M)$. Calabi's conjecture. There \exists a unique Kähler metric $\eta$ on $M$ with $[\eta]=[\omega]$ such that

$$\operatorname{Ric}(\eta)=2\pi\alpha.$$

Calabi established the uniqueness part, while Yau proved existence by solving the associated complex Monge–Ampère equation. Thus the conjecture is solved.