On compactness of K\"ahler metrics with bounded entropy and bounded $L^{2n+1}$ scalar curvature

TL;DR

This paper extends previous results on the regularity of K"ahler metrics by relaxing the boundedness condition on scalar curvature, providing new insights into the compactness of such metrics with bounded entropy and scalar curvature.

Contribution

It introduces a relaxation of the scalar curvature boundedness condition in the study of K"ahler metrics, advancing the understanding of metric compactness.

Findings

Extended regularity results for K"ahler metrics with less restrictive scalar curvature bounds

Established compactness criteria under bounded entropy and scalar curvature

Improved estimates for K"ahler potentials with relaxed assumptions

Abstract

In their seminal work (\cite{CC}, \cite{CC2}), Chen and Cheng proved apriori estimates for the constant scalar curvature metrics on compact K\"ahler manifolds. They also proved $C^{3,\alpha}$ estimate for the potential of the \ka metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this short note is to slightly relax the boundedness condition on the scalar curvature.