Uniform entropy and energy bounds for fully non-linear equations

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-Ampère equations involving sublevel sets, and bypasses the Alexandrov-Bakelman-Pucci maximum principle. In particular, it implies uniform L ∞ bounds for systems coupling a fully non-linear equation to its linearization, generalizing the cscK equation.

Diameter estimates in Kähler geometry

“…[10,16,22,25,31]). However, there has been considerable progress recently in PDE methods for 𝐿 ∞ estimates for fully nonlinear equations [11,12,14]. These new methods turn out to be particularly amenable to geometric estimates, and have been shown to imply some promising estimates for noncollapse [17] and for the Green's function [13].…”