Stability estimates for the complex Monge-Amp\`ere and Hessian equations

TL;DR

This paper introduces a novel proof for stability estimates of complex Monge-Ampère and Hessian equations that avoids pluripotential theory, providing uniform estimates under degenerations of the background metric or class.

Contribution

It presents a new proof technique that yields uniform stability estimates for these equations without relying on pluripotential theory.

Findings

Stability estimates are uniform under degenerations of the background metric.

The method applies to both Monge-Ampère and Hessian equations.

The proof simplifies previous approaches by avoiding pluripotential theory.

Abstract

A new proof for stability estimates for the complex Monge-Amp\`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Amp\`ere equation, and under degenerations to a big class in the case of Hessian equations.