A beautiful phenomenon in Euclidean space is the existence of a 1-parameter family of minimal isometric surfaces connecting the catenoid and the helicoid. They are associate. A well-known fact, is that any two conformal isometric minimal surfaces in a space form are associate. What happens in other 3dimensional manifolds ?In this paper we will discuss the same phenomenon in the product space, M × R, giving a definition of associate minimal immersions. We specialize in the situations M = H 2 , the hyperbolic plane, and M = S 2 , the sphere where surprising facts occur. We will prove some existence and uniqueness results explained in the sequel. We begin with the definition.Let M be a two dimensional Riemannian manifold. Let (x, y, t) be local coordinates in M × R, where z = x + iy are conformal coordinates on M and t ∈ R. Let σ 2 | dz| 2 , be the conformal metric in M, hence ds 2 = σ 2 | dz| 2 + dt 2 is the metric in the product space M×R. Let Ω ⊂ C be a planar simply connected domain, w = u + iv ∈ Ω. We recall that if X : Ω → M × R, w → (h(w), f (w)), w ∈ Ω, is a conformal minimal immersion then h : Ω → (M, σ 2 | dz| 2 ) is a harmonic map. We recall also that for any harmonic map h : Ω ⊂ C → M there exists a related Hopf holomorphic function Q(h). Two conformal immersions X = (h, f ), X * = (h * , f * ) : Ω → H 2 × R are said associate if they are isometric and if the Hopf functions satisfy the relation Q(h * ) = e 2iθ Q(h) for a real number θ. If Q(h * ) = −Q(h) then the two immersions are said conjugate.In this paper we will show that there exist two conformal isometric minimal surfaces in H 2 ×R, with constant Gaussian curvature −1, that are non associate. We will prove also that the vertical cylinder over a planar geodesic in H 2 × R, are the only minimal surfaces with constant Gaussian curvature K ≡ 0.One of our principal results is a uniqueness theorem in H 2 × R, or S 2 × R, showing that the conformal metric and the Hopf function determine a minimal conformal immersion, up to an isometry of ambient space, see Theorem 4. We will derive the existence of the minimal associate family in H 2 × R and S 2 × 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.2000 Mathematics subject classification: 53C42.

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in H 2 × R. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary ∞ is a Jordan curve homologous to zero in ∂ ∞ H 2 × R such that ∞ is contained in a slab between two horizontal circles of ∂ ∞ H 2 × R with width equal to π. We construct vertical minimal graphs in H 2 ×R over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains in H 2 × {0} are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.

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