📝 SummarXiv

Abstract

In this paper, a simplified exposition of the celebrated Aubin-Yau proof for the existence of K\"ahler-Einstein metrics is provided. For the case of a compact K\"ahler manifold with vanishing first Chern class, we improve Yau's celebrated $C^0$ a priori estimate. Instead of relying on the $L^{\infty}$ norm of the K\"ahler potential $F$ as in the original proof, we establish a new uniform bound for the solution to the Monge-Amp\`ere equation that depends only on the $L^{p}$ norm of $e^{F}$. This new approach to the $C^0$ estimate offers an alternative method for establishing uniform bounds for solutions to complex Monge-Amp\`ere equations.