We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.

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

Abstract. We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in H 2 R, S 2 R and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms. Mathematics Subject Classification (2000). 53C30, 53B25.

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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Abstract. In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.

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We prove that a H -surface M in H 2 × R, |H | 1/2, inherits the symmetries of its boundary ∂ M, when ∂ M is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.