Expansion formula for complex Monge-Amp\`ere equation along cone singularities

TL;DR

This paper establishes an asymptotic expansion for solutions to singular complex Monge-Ampère equations related to conical Kähler-Einstein metrics, advancing understanding of their behavior near singularities.

Contribution

It provides the first rigorous proof of the asymptotic expansion for solutions to these singular equations in the context of conical Kähler-Einstein metrics.

Findings

Proved asymptotic expansion of solutions near singularities

Enhanced understanding of conical Kähler-Einstein metrics

Established foundational results for future research

Abstract

In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Amp\`ere equation which arise naturally in the study of the conical K\"ahler-Einstein metric.