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

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.