Uniform entropy and energy bounds for fully non-linear equations

TL;DR

This paper develops uniform energy bounds for fully non-linear equations using entropy-like quantities and auxiliary Monge-Ampère equations, leading to new $L^ infty$ bounds for coupled systems, including generalizations of the cscK equation.

Contribution

It introduces a novel method to obtain uniform energy bounds from entropy bounds, bypassing traditional maximum principles, and extends results to coupled non-linear systems.

Findings

Established uniform $L^ infty$ bounds for fully non-linear equations

Derived bounds for systems coupling non-linear equations with their linearizations

Generalized results to include the cscK equation and similar systems

Abstract

Energy bounds which are uniform in the background metric are obtained from upper bounds for entropy-like quantities. The argument is based on auxiliary Monge-Amp\`ere equations involving sublevel sets, and bypasses the Alexandrov-Bakelman-Pucci maximum principle. In particular, it implies uniform $L^\infty$ bounds for systems coupling a fully non-linear equation to its linearization, generalizing the cscK equation.