Gauss maps of constant mean curvature surfaces in three-dimensional homogeneous spaces

TL;DR

This paper classifies constant mean curvature surfaces in 3D homogeneous spaces based on the harmonicity properties of their Gauss maps, extending classical results from space forms to more general ambient spaces.

Contribution

It provides a complete classification of CMC surfaces with vertically harmonic Gauss maps in all 3D homogeneous spaces, generalizing known results from space forms.

Findings

Classification of CMC surfaces with vertically harmonic Gauss maps

Extension of harmonicity results from space forms to homogeneous spaces

Identification of conditions for Gauss map harmonicity in homogeneous spaces

Abstract

It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for surfaces in an arbitrary 3-dimensional ambient space. In this paper we study this problem for surfaces in an important and very natural family of 3-dimensional ambient spaces, namely homogeneous spaces. In particular, we obtain a full classification of constant mean curvature surfaces, whose Gauss map satisfies the more mild condition of vertical harmonicity, in all 3-dimensional homogeneous spaces.