Embeddedness of spheres in homogeneous three-manifolds

TL;DR

This paper proves that certain immersed surfaces with a diffeomorphic Gauss map in specific three-dimensional Lie groups are actually embedded spheres, and applies this to show constant mean curvature spheres of index one are embedded.

Contribution

It establishes that surfaces with a diffeomorphic left invariant Gauss map in homogeneous three-manifolds are embedded spheres, extending classical results to new geometric contexts.

Findings

Surfaces with a diffeomorphic Gauss map are embedded spheres.

Constant mean curvature spheres of index one are embedded.

Results apply to metric Lie groups with algebraic open book decompositions.

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.