Singular scalar curvature equations

TL;DR

This paper develops estimates for scalar curvature equations of singular metrics with cone angles on Kähler manifolds, advancing understanding of singular constant scalar curvature and Kähler-Einstein metrics.

Contribution

It introduces new estimates for scalar curvature equations of singular metrics and applies them to singular constant scalar curvature and Kähler-Einstein metrics.

Findings

Established Laplacian estimates for degenerate Kähler metrics

Derived estimates for scalar curvature of singular metrics with cone angles

Applied estimates to singular Kähler-Einstein metrics

Abstract

We develop estimates for the equation of scalar curvature of singular metrics with cone angle $\beta>1$, in a big and semi-positive cohomology class on a K\"ahler manifold. We further derive the Laplacian estimate for the scalar curvature equation of degenerate K\"ahler metrics. We then have several applications of these estimates on the singular constant scalar curvature K\"ahler metrics, which also include the singular K\"ahler-Einstein metrics.