Generalization of Finite Entropy Measures in K\"{a}hler Geometry

TL;DR

This paper generalizes finite entropy measures in K"ahler geometry, introduces finite p-entropy, and proves stability of the complex Monge-Ampère equation on compact K"ahler manifolds.

Contribution

It extends the concept of finite entropy to p-entropy in K"ahler geometry and establishes a stability result for the Monge-Ampère equation.

Findings

Finite p-entropy measures are included in specific energy classes.

A stability result for the complex Monge-Ampère equation is proved.

The work builds on Darvas's Finsler metric on energy classes.

Abstract

In this paper, we extend the concept of finite entropy measures in K\"ahler geometry. We define the finite $p$-entropy related to $\omega$-plurisubharmonic functions and demonstrate their inclusion in an appropriate energy class. Our study is anchored in the analysis of finite entropy measures on compact K\"ahler manifolds, drawing inspiration from fundamental works of Di Nezza, Guedj, and Lu. Utilizing a celebrated result by Darvas on the existence of a Finsler metric on the energy classes, we conclude this paper with a stability result for the complex Monge-Amp\`ere equation.