Embeddedness of spheres in homogeneous three-manifolds

Abstract

Let $X$ denote a metric Lie group diffeomorphic to ${\mathbb{R}}^3$ that admits an algebraic open book decomposition. In this paper we prove that if $\Sigma$ is an immersed surface in $X$ whose left invariant Gauss map is a diffeomorphism onto ${\mathbb{S}}^2$, then $\Sigma$ is an embedded sphere. As a consequence, we deduce that any constant mean curvature sphere of index one in $X$ is embedded.

12 pages, 2 figures