Cylinders as Left Invariant CMC Surfaces in $\operatorname{Sol}_3$ and $E(\kappa,\tau)$-Spaces Diffeomorphic to $\mathbb{R}^3$

TL;DR

This paper proves the existence of certain constant mean curvature cylinders in specific three-dimensional homogeneous manifolds, including new examples in Sol_3 and related spaces, using geometric and Lie group methods.

Contribution

It provides a geometric proof for the existence of CMC cylinders in homogeneous 3-manifolds and constructs new properly embedded CMC annuli in Sol_3 and similar spaces.

Findings

Existence of CMC cylinders in homogeneous 3-manifolds.

Construction of new properly embedded CMC annuli in Sol_3.

Examples of cylinders generated by non-embedded curves.

Abstract

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature $H>H(X)$ in certain simply connected homogeneous three-manifolds $X$ diffeomorphic to $\mathbb{R}^3$, which always admit a Lie group structure. Here, $H(X)$ denotes the critical value for which constant mean curvature spheres in $X$ exist. Our cylinders are generated by a simple closed curve under a one-parameter group of isometries, induced by left translations along certain geodesics. In the spaces $\operatorname{Sol}_3$ and $\widetilde{\operatorname{PSL}}_2(\mathbb{R})$ we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in $\operatorname{Sol}_3$ generated by non-embedded simple closed curves.