# Hermitian curvature flow on complex locally homogeneous surfaces

## TL;DR

This paper investigates the Hermitian curvature flow on complex surfaces, characterizing long-term behavior, providing the first example of finite-time singularity, and analyzing the Gromov-Hausdorff limits of solutions.

## Contribution

It offers a detailed case-by-case analysis of the flow on complex model geometries, including the first example of finite-time singularity in this context.

## Key findings

- Characterization of long-time behavior of the flow
- First example of finite-time singularity in Hermitian curvature flow
- Gromov-Hausdorff limits of solutions after normalization

## Abstract

We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-K\"ahler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov-Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry.