Graphs and multi-graphs in homogeneous 3-manifolds with isometry groups of dimension 4

Abstract

Abstract. We study the existence of multi-graphs which are immersed in E 3 (κ, τ ), having constant mean curvature H, where E 3 (κ, τ ) is a homogeneous, simply connected 3-manifold whose isometry group has dimension 4.

Generalized Calabi correspondence and complete spacelike surfaces

“…Proof. We will use a classical argument [14,19,29,31] originally due to Heinz to obtain item (a). Aiming at a contradiction, suppose that such an entire graph exists, and assume first that H > 0 (the case H < 0 will be treated later).…”

The half space property for cmc 1/2 graphs in $$\mathbb {E}(-1,\tau )$$ E ( - 1 , τ )

“…where the mean curvature is computed with respect to the upward pointing normal (see (14) in Appendix A for a proof of (2)). If we use the coordinates given by the model, a section σ can be identified to a function (i.e.…”

“…If σ is a section defined on the whole H 2 and its mean curvature is constant H 0 , it is known that this mean curvature is less than 1/2 (see [14]). In fact, we are interested in entire graph with mean curvature 1/2; so we consider sections σ defined on the whole H 2 which are solution of div H 2 Gσ…”

“…where the mean curvature is computed with respect to the upward pointing normal (see (14) in Appendix A for a proof of ( 2)).…”