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

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.