K\"ahler families of Green's functions

TL;DR

This paper extends existing uniform estimates for Green's functions in K"ahler geometry by relaxing assumptions and allowing complex structures to vary, with applications to canonical K"ahler metrics.

Contribution

It generalizes previous results on Green's functions by removing assumptions and considering variable complex structures in K"ahler families.

Findings

Broadened the scope of Green's function estimates in K"ahler geometry.

Allowed complex structures to vary within families.

Applied results to canonical K"ahler metrics.

Abstract

In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green's functions associated with certain K\"ahler metrics. In this note, we broaden the scope of their techniques by removing one of their assumptions and allowing the complex structure to vary. We apply our results to various families of canonical K\"ahler metrics.