A "boundedness implies convergence" principle and its applications to collapsing estimates in K\"ahler geometry

TL;DR

This paper introduces a general principle linking boundedness to convergence for evolving Riemannian metrics and applies it to collapsing Calabi-Yau metrics and K"ahler-Ricci flows, demonstrating convergence in these geometric contexts.

Contribution

It presents a new 'boundedness implies convergence' principle and applies it to complex geometric flows, providing new convergence results for collapsing metrics.

Findings

Established a general convergence principle for Riemannian metrics.

Proved convergence of collapsing Calabi-Yau metrics.

Demonstrated convergence of normalized K"ahler-Ricci flows on torus fibered models.

Abstract

We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized K\"ahler-Ricci flows on torus fibered minimal models to obtain convergence results.