Conjugate Plateau constructions in product spaces

TL;DR

This survey explores the geometric theory of conjugate minimal and constant mean curvature surfaces in homogeneous 3-manifolds, reviewing existing results, constructions, and providing numerical visualizations.

Contribution

It consolidates known results on conjugate surfaces, revisits constructions in product spaces, and includes numerical visualizations, offering a comprehensive geometric perspective.

Findings

Collected and analyzed existing strategies for conjugate surface analysis.

Revisited constructions of constant mean curvature surfaces in product spaces.

Provided numerical visualizations of surfaces using Surface Evolver.

Abstract

This survey paper investigates, from a purely geometric point of view, Daniel's isometric conjugation between minimal and constant mean curvature surfaces immersed in homogeneous Riemannian three-manifolds with isometry group of dimension four. On the one hand, we collect the results and strategies in the literature that have been developed so far to deal with the analysis of conjugate surfaces and their embeddedness. On the other hand, we revisit some constructions of constant mean curvature surfaces in the homogeneous product spaces $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{R}^3$ having different topologies and geometric properties depending on the value of the mean curvature. Finally, we also provide some numerical pictures using Surface Evolver.