SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces

Totally umbilic surfaces in homogeneous 3-manifolds

“…More generally many works are devoted to studying the geometry of surfaces in homogeneous 3-manifolds. See for example [14], [6], [7], [17], [15], [12], [13], [11], [9], [4], [2], [10], [5] and [8].…”

Invariant surfaces with constant mean curvature in $${\mathbb{H}}^2 \times {\mathbb{R}}$$

“…Before starting the study of (2) we make some elementary remarks (see [9]). (2). Thus the solutions of (2) are characterized by the equatioṅ Table 6).…”

“…2 When a = 1 and b = 0 we have that v(u) = k F(u, k 2 ). Note that it is enough to study the case k > 0.…”

“…• For k = 0, as lim u→0 + sin σ = −H and lim u→π − sin σ = H , we have that: 2 The elliptic integral of the first kind is defined by: -if H > 1, the curve γ does not reach the lines u = 0 and u = π; -if H = 1, the curve γ tends asymptotically to the lines u = 0 and u = π; -if H < 1, the curve γ tends to the lines u = 0 and u = π with an angle σ = arcsin(−H ) and σ = arcsin(H ), respectively.…”

“…For surfaces in a three-dimensional manifold the unique interesting (not trivial) subgroups of isometries are the one-parameter subgroups and, in this case, any invariant surface can be rendered as the orbit of a curve (the profile curve) by the action of the subgroup. The study of invariant surfaces in the three-dimensional Thurston's geometries has been initiated by Caddeo, Piu and Ratto in [2], where they characterized the SO(2)-invariant CMC surfaces in a three-dimensional homogenous space, and by Tomter, in [13], for the Heisenberg group H 3 . Also, in [3], the authors described the profile curves of the SO(2)-invariant surfaces with constant Gauss curvature in the Heisenberg group H 3 .…”