The uniqueness statement in the setting of (i) is due to Calabi. The much harder existence part was proved by Yau [Yau78], using PDE techniques. The combined result is often called the Calabi-Yau Theorem.

The general setting of (ii)–(iii) was treated by Kołodziej [Koł98, Koł03] who used methods of pluripotential theory together with a nontrivial reduction to Yau’s result. Guedj and Zeriahi [GZ07] more generally established the existence of solutions of $\operatorname{MA}(\phi)=\mu$ for positive measures $\mu$ (of mass $(L^{n})$) that do not put mass on pluripolar sets. In this generality, the metrics $\phi$ are no longer continuous but rather lie in a suitable energy class, modeled upon work by Cegrell [Ceg98]. Dinew [Din09], improving upon an earlier result by Błocki [Bło03], proved the corresponding uniqueness theorem. All these existence and uniqueness results are furthermore valid (in a suitable formulation) in the transcendental case, when $(X,\omega)$ is a Kähler manifold.

The complex Monge-Ampère equation is of fundamental importance to complex geometry. For example, it implies that every compact complex manifold with vanishing first Chern class (such manifolds are now called Calabi-Yau manifolds) admit a Ricci flat metric in any given Kähler class. The complex Monge-Ampère equation also plays a key role in recent work on the space of Kähler metrics.