# Parallel mean curvature surfaces in four-dimensional homogeneous spaces

## TL;DR

This paper reviews classification results for surfaces with parallel mean curvature in four-dimensional homogeneous spaces, emphasizing holomorphic differentials and special cases like spheres, summarizing recent advances in the field.

## Contribution

It provides a unified framework for understanding parallel mean curvature surfaces in various four-manifolds, highlighting the role of holomorphic quadratic differentials and recent progress.

## Key findings

- Classification results for such surfaces in different spaces
- Existence of holomorphic quadratic differentials on these surfaces
- Detailed analysis of spheres with parallel mean curvature

## Abstract

We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for this problem, with special attention to the existence of holomorphic quadratic differentials on such surfaces. The case of spheres with parallel mean curvature is also explained in detail, as well as the state-of-the-art advances in the general problem.