# The Chern-Ricci flow on primary Hopf surfaces

## TL;DR

This paper investigates the behavior of the Chern-Ricci flow on primary Hopf surfaces, revealing finite-time volume collapsing singularities and supporting conjectures about their Gromov-Hausdorff limits, with new uniform estimates established.

## Contribution

It provides the first detailed analysis of the Chern-Ricci flow on primary Hopf surfaces of class 1, including uniform bounds and singularity behavior, extending previous results beyond simple examples.

## Key findings

- Solutions reach finite-time volume collapsing singularity.
- Metric tensor has a uniform upper bound.
- Established uniform $C^{1+eta}$ estimates for the potential.

## Abstract

The Hopf surfaces provide a family of minimal non-K\"ahler surfaces of class VII on which little is known about the Chern-Ricci flow. We use a construction of Gauduchon-Ornea for locally conformally K\"ahler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern-Ricci flow. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the Gromov-Hausdorff limit is isometric to a round $S^1$. Uniform $C^{1+\beta}$ estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.