Diameter estimates in K\"ahler geometry II: removing the small degeneracy assumption

TL;DR

This paper removes the small degeneracy assumption from previous Kähler geometry estimates, establishing diameter, Green's functions, and Sobolev inequalities under entropy bounds without degeneracy constraints.

Contribution

It provides a technical improvement that extends geometric estimates in Kähler geometry to cases without the small degeneracy assumption.

Findings

Diameter estimates under entropy bounds

Green's function bounds without degeneracy

Sobolev inequalities without small degeneracy assumption

Abstract

In this short note, we remove the small degeneracy assumption in our earlier works [10, 11]. This is achieved by a technical improvement of Corollary 5.1 in [10]. As a consequence, we establish the same geometric estimates for diameter, Green's functions and Sobolev inequalities under an entropy bound for the K\"ahler metrics, without any small degeneracy assumption.