Jun-ichi Inoguchi, Joeri Van der Veken
Published 2010-09-01, updated 2011-04-15Version 2
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.
New examples of constant mean curvature surfaces in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$
Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods
Constant Mean Curvature Surfaces in $\mathbb{M}^2(c)\times\mathbb{R}$ and Finite Total Curvature
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