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

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Abstract. We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil 3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil 3 is entire. Also, the sister surface of an entire minimal graph in Nil 3 is an entire constant mean curvature annuli in H 2 × R.

Dedicated to David Hoffman and Bill Meeks on the occasion of their 60th birthdaysFor a complete embedded surface with compact boundary and constant mean curvature 1 2 in H 2 × R lying on one side of a horocylinder, we prove an analogue of the Hoffman-Meeks half-space theorem. As an application, we show that a complete immersed surface of constant mean curvature 1 2 which is transverse to the vertical killing field must be an entire graph. Moreover, to each holomorphic quadratic differential on the unit disk or C we can associate an entire graph of constant mean curvature 1 2 .

We construct and classify minimal surfaces foliated by horizontal curves of constant curvature in ‫ވ‬ 2 ‫,ޒ×‬ ‫ޒ‬ 2 ‫ޒ×‬ and ‫ޓ‬ 2 ‫.ޒ×‬ The main tool is the existence of a Shiffman Jacobi field; such fields characterize the property of being foliated by circles in these product manifolds.

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In this note, we explain how a mistake made in [2] can be corrected. Actually this mistake appears in the proof of Proposition 8 (the second maximum principle) and was brought to our attention by A. Song [4]. Let us notice that unfortunately, we did not find an alternative proof of this proposition but we found an alternative proposition. The new proposition does not change the subsequent applications we made of the original proposition. At the end of the note we explain which modifications should be done where the original Proposition 8 is applied.The difference with the original Proposition 8 is that here we have to assume a control on the index of the minimal surface Σ.The new proposition. Let (T, ds 2 T ) be a flat 2 torus of diameter 1 and consider M = T×[− 1 2 , +∞) endowed with the metric g = e −2h(t) Λ 2 ds 2 T +dt 2 ; here Λ is a positive constant and h satisfies the following assumptions:•The main example is h(t) = t. We denote by T s = T × {s}.Proposition 1. Let i 0 ∈ N. There is a Λ 0 such that the following is true. For any Λ ≤ Λ 0 and any compact embedded minimal surface Σ ⊂ M with ∂Σ ⊂ T − 1 2 and index less than i 0 , we have Σ ⊂ T × [− 1 2 , 0]Proof. If the proposition is not true there is a sequence Λ n → 0 and Σ n a minimal surface in M with index less than i 0 , boundary in T − 1 2 and the maximum of the function t on Σ n is t n > 0.In order to study this sequence, we translate in the t direction by −t n and make a homothety by λ n = e h(tn) Λn → +∞. We thus obtain a compact minimal surface S n in T × [−( 1 2 + t n )λ n , 0] endowed with the metric g n = e −2hn(t) ds 2 T + dt 2 (where h n (t) = h( t λn + t n ) − h(t n )) with boundary in T −( 1 2 +tn)λn and containing a point in T 0 . We notice that h n → 0 uniformly on any compact. So the ambient space smoothly converges to X = T × R − endowed with the flat metric ds 2 T × dt 2 . Let ε be positive. Since S n has index at most i 0 , there is a set E n (ε) of at most i 0 points in S n such that, for any p ∈ X with d(p, E n (ε)) > 2ε, 1

2 The Costa-Hoffman-Meeks' family of minimal surfaces C. Costa [2], [3] and later on D. Hoffman and W. H. Meeks [7], [8] have described, for k ≥ 1, a properly embedded minimal surface of genus k with three ends. More precisely, for each k ≥ 1, there exists M k a complete properly embedded minimal surface of genus k and three ends which, after suitable rotation and translation, enjoys the following properties :