Diameter estimates in K\"ahler geometry

TL;DR

This paper derives diameter bounds for K"ahler metrics using entropy bounds without requiring Ricci curvature lower bounds, employing advanced PDE techniques related to the Monge-Ampère equation.

Contribution

It introduces new PDE methods that handle degeneracies in volume forms, enabling diameter estimates in broader K"ahler geometric contexts.

Findings

Diameter bounds for K"ahler metrics under entropy conditions

Application to long-time solutions of K"ahler-Ricci flow

Diameter estimates for Calabi-Yau fibrations

Abstract

Diameter estimates for K\"ahler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $L^\infty$ estimates for the Monge-Amp\`ere equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time solutions of the K\"ahler-Ricci flow and finite-time solutions when the limiting class is big, as well as for special fibrations of Calabi-Yau manifolds.