Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in H n+1 , by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on S n for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result in a more general framework. Specifically, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in H n+1 is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on S n and the theory of hypersurfaces with prescribed hyperbolic Gauss map in H n+1. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals W(κ 1 ,. .. , κ n), the only compact immersed hypersurfaces in H n+1 on which W is constant are round spheres.

We introduce a new approach to the local study of maximal surfaces in Lorentz-Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Björling-type problem in Lorentz-Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces. † Partially supported by Dirección General de Investigación (MCYT) BFM2001-2871 and Consejería de Educación y Universidades (CARM) Programa Séneca, PI-3/00854/FS/01.

We introduce a hyperbolic Gauss map into the Poincaré disk for any surface in H 2 × R with regular vertical projection, and prove that if the surface has constant mean curvature H = 1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere conformal harmonic map from an open simply connected Riemann surface Σ into the Poincaré disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on Σ can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H = 1/2 in H 2 × R. A similar result applies to minimal surfaces in the Heisenberg group Nil 3 . Finally, we classify all complete minimal vertical graphs in H 2 × R. SetupIn this preliminary section we will describe some general facts that will be used in the study of both minimal surfaces and surfaces with H = 1/2 in H 2 × R. First, we will analyze the structure and compatibility equations of an immersed surface in H 2 × R in terms of a conformal parameter for its first fundamental form. Subsequently, we will make some comments regarding harmonic maps into the hyperbolic plane H 2 and their relation with spacelike CMC surfaces in the Lorentz-Minkowski 3-space L 3 .

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Abstract. We classify the entire minimal vertical graphs in the Heisenberg group Nil 3 endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil 3 , is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil 3 .