Immersed hyperbolic and parabolic screw motion surfaces in the space $$\widetilde{\textit{PSL}}_{2}(\mathbb {R},\tau )$$ PSL ~ 2 ( R , τ )

Graph Surfaces Invariant by Parabolic screw Motions with Constant Curvature in $ \: \mathbb H^2 \times \mathbb R$

“…Parabolic and hyperbolic screw motion surfaces in H 2 × R were studied in [4,18]. For the definition of parabolic screw motion surfaces we follow [4].…”

“…Parabolic and hyperbolic screw motion surfaces in H 2 × R were studied in [4,18]. For the definition of parabolic screw motion surfaces we follow [4]. We will use helicoidal-type isometries in H 2 × R which are the composition of isometries of H 2 together with vertical translation in a proportional way.…”

“…In [12,17], H. Rosenberg and W. Meeks studied minimal surfaces in M 2 × R, where M 2 is a rounded sphere, a complete Riemannian surface with a metric of non-negative curvature, or M 2 = H 2 , the hyperbolic plane. Since then, there has been a rapid growing interest in minimal surfaces and surfaces with constant mean curvature in H 2 × R and S 2 × R, see for instance [4,5,9,13,14,15,18,19,20]. Also, surfaces in H 2 × R and S 2 × R with constant Gaussian curvature or constant extrinsic curvature have attracted many attention in the recent years, [1,2,3,6,7,16].…”

“…R. Sa Earp and E. Toubiana [19] obtained an explicit two parameter family of complete, embedded, simply connected, minimal screw motion surfaces in H 2 × R with pitch ℓ, and for ℓ = 1 each such surface has Gaussian curvature K = −1. In [18] R. Sa Earp studied complete minimal and surfaces with constant mean curvature invariant either by parabolic or by hyperbolic screw motions in H 2 × R. Later, Q. Cui and et al [4] studied the geometric behaviors of hyperbolic and parabolic screw motions surfaces immersed in P SL 2 (R, τ ) with having constant mean curvature, where P SL 2 (R, τ ) is a homogeneous simply connected 3-manifold having isometry group of dimension 4.…”