Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

TL;DR

This paper establishes convergence results for the normalized Chern-Ricci flow on complex minimal surfaces of general type, removing previous assumptions and confirming a conjecture in complex dimension two.

Contribution

It proves uniform geometric estimates and Gromov-Hausdorff convergence for the flow without local Kahler assumptions, confirming the Tosatti-Weinkove conjecture.

Findings

Uniform diameter estimates and volume non-collapsing established.

Gromov-Hausdorff convergence proven for the flow.

Removal of local Kahler assumption near the null locus.

Abstract

We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.