On the constant scalar curvature K\"ahler metrics, apriori estimates

TL;DR

This paper establishes a priori estimates for constant scalar curvature K"ahler metrics on compact manifolds, linking higher derivatives to bounds on the K"ahler potential, with implications for local estimates.

Contribution

It provides new a priori estimates relating higher derivatives to the K"ahler potential bounds, including local versions of these estimates.

Findings

Higher order derivatives are controlled by $C^0$ bounds.

Local estimates are developed for potential independent analysis.

Results aid in understanding regularity of K"ahler metrics.

Abstract

In this paper, we derive apriori estimates for constant scalar curvature K\"ahler metrics on a compact K\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\"ahler potential. We also discuss some local versions of these estimates which can be of independent interest.