Constant mean curvature surfaces in 3-dimensional Thurston geometries

TL;DR

This survey reviews the global theory of constant mean curvature surfaces across Thurston's eight 3-dimensional geometries, focusing on classification problems and open questions in the field.

Contribution

It provides a comprehensive overview of the classification of compact and entire CMC surfaces in Thurston geometries, highlighting key open problems and recent developments.

Findings

Classification of compact CMC surfaces in various geometries

Existence results for entire CMC graphs

Identification of open problems in the field

Abstract

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2 \times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented.