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arXiv · 2510.00609

A Simplification of the Aubin-Yau Proof and an Alternative $C^{0}$ Estimate for the Monge-Amp\`ere Equation on Calabi-Yau Manifolds

Junyu Pan
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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, the analysis presents an alternative formulation of the $C^0$ a priori estimate. Instead of relying on the $L^{\infty}$ norm of the K\"ahler potential $F$ as in the original proof, a different uniform bound for the solution to the Monge-Amp\`ere equation that depends only on the $L^{p}$ norm of $e^{F}$ is established. This estimate has a stronger version established by Ko{\l}odziej in 1998.

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Junyu Pan. 2025-10-01. A Simplification of the Aubin-Yau Proof and an Alternative $C^{0}$ Estimate for the Monge-Amp\`ere Equation on Calabi-Yau Manifolds. https://arxiv.org/abs/2510.00609

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