A Simplification of the Aubin-Yau Proof and an Alternative $C^{0}$ Estimate for the Monge-Ampère Equation on Calabi-Yau Manifolds
Junyu Pan
TL;DR
The work supplies a streamlined presentation of the Aubin–Yau framework for Kähler–Einstein metrics on compact Kähler manifolds by recasting the problem as a complex Monge–Ampère equation and applying the continuity method. It introduces an alternative $C^{0}$ a priori estimate in the Calabi–Yau case based on the $L^{p}$ norm of $e^{F}$, with Kołodziej’s strengthened results highlighted. The discussion covers the negative, zero, and positive first Chern class scenarios, clarifying the analytic steps (a priori estimates and bootstrapping) and the role of stability notions in the Fano case. Overall, the paper clarifies the classical proofs, provides a more accessible route to the Calabi–Yau metric, and connects $C^{0}$ control to stability concepts in Fano geometry.
Abstract
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 $L^{\infty}$ 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.