A Simplification of the Aubin-Yau Proof and an Alternative $C^0$ Estimate for the Monge-Ampère Equation on Calabi-Yau Manifolds

In this paper, a simplified exposition of the celebrated Aubin-Yau proof for the existence of Kähler-Einstein metrics is provided. For the case of a compact Kähler manifold with vanishing first Chern class, the analysis presents an alternative formulation of the $C^0$ a priori estimate. Instead of relying on the $L^{\mathrm{\infty }}$ norm of the Kähler potential $F$ as in the original proof, a different uniform bound for the solution to the Monge-Ampère equation that depends only on the $L^p$ norm of $e^F$ is established. This estimate has a stronger version established by Kołodziej in 1998.