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

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 ${\mathrm{Sol}}_3$ and ${\widetilde{\mathrm{PSL}}}_2(\mathbb{R})$ we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in ${\mathrm{Sol}}_3$ generated by non-embedded simple closed curves.

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References in corpus (1)

• New examples of constant mean curvature surfaces in ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$

Cited by in corpus (2)

• Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(\kappa ,\tau )$-spaces
• Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds