Parabolic and Hyperbolic Screw Motion Surfaces in ℍ²×ℝ

In this paper we find many families in the product space ℍ 2 ×ℝ of complete embedded, simply connected, minimal and surfaces with constant mean curvature H such that | H |≤1/2. We study complete surfaces invariant either by parabolic or by hyperbolic screw motions. We study the notion of isometric associate immersions. We exhibit an explicit formula for a Scherk-type minimal surface. We give a one-parameter family of entire vertical graphs of mean curvature 1/2. We prove a generalized Bour lemma that can be applied to ℍ 2 ×ℝ,𝕊 2 ×ℝ and to Heisenberg’s space to produce a family of screw motion surfaces isometric to a given one.