Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

TL;DR

This paper proves convergence and uniqueness results for the Chern-Ricci flow on certain Hermitian manifolds, extending K"ahler techniques to the non-K"ahler setting and partially resolving a conjecture.

Contribution

It establishes uniform estimates and Gromov-Hausdorff convergence for the flow, introduces Perelman's reduced length to the Hermitian context, and proves limit uniqueness.

Findings

Proved subsequential Gromov-Hausdorff convergence of the flow.

Established a uniform Chern scalar curvature bound.

Demonstrated the uniqueness of the limit space when the manifold is K"ahler.

Abstract

We establish uniform diameter estimates and volume non-collapsing estimates for the Chern-Ricci flow on smooth Hermitian minimal models of general type, assuming the initial metric is K\"ahler in a neighborhood of the null locus of the canonical bundle. This yields subsequential Gromov-Hausdorff convergence, partially resolving a conjecture of Tosatti and Weinkove. When the underlying manifold is K\"ahler, we further prove the uniqueness of the limit space. Analytically, we overcome the difficulties posed by non-K\"ahler torsion in the Green's formula by exploiting our local K\"ahler assumption, successfully adapting recent estimates of K\"ahler Green's function to the Hermitian setting. To prove the uniqueness of the limit, we introduce Perelman's reduced length to the Chern-Ricci flow. By establishing a uniform Chern scalar curvature bound and an almost monotonicity formula for the reduced volume, we deduce an almost-avoidance principle for the singular set, allowing us to effectively compare the flow distance with the canonical limit distance.