Quantitative stability for the complex Monge-Ampere equations

TL;DR

This paper establishes quantitative stability estimates for solutions to complex Monge-Ampère equations, especially under variations in cohomology classes and singularities, contributing to the understanding of degenerating Kähler-Einstein metrics.

Contribution

It introduces new stability estimates using pluripotential theory, advancing the analysis of complex Monge-Ampère equations with varying parameters.

Findings

Quantitative stability estimates derived for solutions under parameter variations

Application to degeneration of Kähler-Einstein metrics

Method based on pluripotential theory in finite energy spaces

Abstract

We prove several quantitative stability estimates for solutions of complex Monge-Ampere equations when both the cohomology class and the prescribed singularity vary. In a broad sense, our results fit well into the study of degeneration of families of Kaehler-Einstein metrics. The key mechanism in our method is the pluripotential theory in the space of potentials of finite lower energy.