On relative $L^\infty$ estimate for complex Monge-Amp\`ere equations

TL;DR

This paper establishes a new relative $L^ Infty$ estimate for complex Monge-Amp e e equations on K"ahler manifolds, unifying previous estimates and improving stability and regularity results.

Contribution

It introduces a unified approach to $L^ Infty$ estimates for complex Monge-Amp e e equations, enhancing previous stability and regularity bounds using PDE techniques.

Findings

Unified $L^ Infty$ estimate for Monge-Amp e e equations

Improved stability and modulus of continuity results

Enhanced estimates for Green's functions

Abstract

We prove a relative $L^\infty$ estimate for a class of complex Monge-Amp\`ere type equations on K\"ahler manifolds. It provides a unified approach to Tundinger type estimate and uniform estimate. It also improves the previous results about modulus of continuity, stability estimates, and $W^{1,1}$-estimates of Green's functions. The argument is based on the PDE method developed by Guo-Phong-Tong and constructing appropriate comparison metrics from entropy bound.