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We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in R³. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a "Lagrangian" graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation. [ABSTRACT FROM AUTHOR]

We prove that any complete, uniformly elliptic Weingarten surface in Euclidean 3-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. To obtain these results, we prove that planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form. [ABSTRACT FROM AUTHOR]

Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups G acting properly nonelementarily by isometries on a proper CAT(0) space X. More precisely, proper proximality holds in the presence of rank one isometries or when X is a locally thick affine building with a minimal G-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary CAT(0) cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a 2-dimensional piecewise Euclidean CAT(0) simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana, and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space. [ABSTRACT FROM AUTHOR]

This article is concerned with the interplay between the theory of smooth valuation on manifolds and Riemannian geometry. We confirm the angularity conjecture formulated by A. Bernig, J. H. G. Fu, and G. Solanes which sheds new light on the geometric meaning of the Lipschitz-Killing valuations. The proof relies on a complete classification of translation-invariant angular curvature measures on Rn, a result of independent interest. [ABSTRACT FROM AUTHOR]

We consider the following variational problem: among all curves in Rn of fixed length with prescribed end points and prescribed tangents at the end points, minimise the L∞-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of differential equations. Furthermore, we have a lot of information about the structure of solutions, which allows a classification. [ABSTRACT FROM AUTHOR]

In this paper we construct families of bounded domains Ωε and solutions uε of such that, for any integer k ≥ 2, uε admits at least k maximum points for small enough. The domain Ωε is "not far" to be convex in the sense that it is starshaped, the curvature of ∂Ωε vanishes at exactly two points and the minimum of the curvature of ∂Ωε goes to 0 as ε→0. [ABSTRACT FROM AUTHOR]

In this paper we show the existence of weak solutions w: M R of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of w and for the mean curvature of its level sets, which are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the p-Laplace equation, and relies on new gradient and decay estimates for p-harmonic capacity potentials, notably for the kernel Gp of Δp. These bounds, stable as p1, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of w. [ABSTRACT FROM AUTHOR]

We establish structure theorems for a smooth projective variety X with semi-positive holomorphic sectional curvature. We first prove that X is rationally connected ifX has no truly flat tangent vectors at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive). This result solves Yau's conjecture on positive holomorphic sectional curvature in a strong form. Moreover, we prove that X admits a locally trivial morphism X =Y such that the fiber F is rationally connected and the image Y has a finitetale cover A=Y by an abelian variety A. We also show that the universal cover of X is biholomorphic and isometric to the product Cm×F of the complex Euclidean space Cm with the flat metric and the rationally connected fiber F with the induced Kahler metric. Our structure theorem is a natural generalization of the structure theorem established by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. [ABSTRACT FROM AUTHOR]

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in Sn with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in S³, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in S², we completely classify ancient flows of bounded length: they are steady or shrinking circles. [ABSTRACT FROM AUTHOR]

We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume growths under an upper curvature bound. [ABSTRACT FROM AUTHOR]

This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns. [ABSTRACT FROM AUTHOR]

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X ⊂ R²m-1, we find geometric significance in the function MX(R) = mag(R·X). The function MX extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit R → ∞, MX admits an asymptotic expansion. The three leading terms of MX at R =+∞ are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex X the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients. [ABSTRACT FROM AUTHOR]

In this paper we study the class of compact K¨ahler manifolds with Ric⊥ >0. First we illustrate examples of K¨ahler manifolds with Ric⊥ > 0 on K¨ahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of K¨ahler manifolds which admit metrics of Ric⊥ > 0. Secondly we prove some (algebraic) geometric consequences of the condition Ric⊥ > 0 to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four. [ABSTRACT FROM AUTHOR]

On a bounded strictly pseudoconvex domain in Cn, n>1, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in C² which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. In particular we observe that, up to biholomorphism, the unit ball in C² is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. We further show that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Finally, we give a generalization of the recent result of the second author that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR 3-sphere. [ABSTRACT FROM AUTHOR]

We obtain compact orientable embedded surfaces with constant mean curvature 0

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there are no solutions for the curvature equation u+eu = 8pnd0 on Et, n N, where Et is a flat rectangular torus and d0 is the Dirac measure at the lattice points. This confirms a conjecture of Lin and Wang and also improves a result of Eremenko and Gabrielov. The nonexistence is a delicate problem because the equation always has solutions if 8pn in the RHS is replaced by 2p with 0 < 4N. Geometrically, our result implies that a rectangular torus Et admits a metric with curvature +1 acquiring a conic singularity at the lattice points with angle 2pa if and only if a is not an odd integer. Unexpectedly, our proof of the nonexistence result is to apply the spectral theory of finite-gap potential, or equivalently the algebro-geometric solutions of stationary KdV hierarchy equations. Indeed, our proof can also yield a sharp nonexistence result for the curvature equation with singular sources at three half periods and the lattice points. [ABSTRACT FROM AUTHOR]

We present new free-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of oneparameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere in the complex projective space, endowed with an arbitrary Riemannian metric, whose onecycle length is bounded in terms of the volume (or diameter) of the manifold. This is the first estimate on sweepout volume in higher dimension without curvature assumption. We also give a detailed account of the situation for compact Riemannian surfaces with or without boundary, in relation with questions raised by P. Buser and L. Guth. [ABSTRACT FROM AUTHOR]

We prove that any complete immersed two-sided mean convex translating soliton Σ ⊂ R³ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in R³ is the axisymmetric "bowl soliton". We also show that if the mean curvature of Σ tends to zero at infinity, then Σ can be represented as an entire graph and so is the "bowl soliton". Finally we classify the asymptotic behavior of all locally strictly convex graphical translating solitons defined over strip regions. [ABSTRACT FROM AUTHOR]

The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k = n-1. We also discuss some applications of our estimates. [ABSTRACT FROM AUTHOR]

In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are geodesic balls in Sm, punctured balls and annular domains. Our results extend those of Escobar when m≥ 3, and Hang-Wang and Jiménez when m= 2. [ABSTRACT FROM AUTHOR]

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders. [ABSTRACT FROM AUTHOR]

We consider properly immersed finite topology minimal surfaces Σ in complete finite volume hyperbolic 3-manifolds N, and in M ×S¹, where M is a complete hyperbolic surface of finite area. We prove Σ has finite total curvature equal to 2π times the Euler characteristic χ(Σ) of Σ, and we describe the geometry of the ends of Σ. [ABSTRACT FROM AUTHOR]

Let g(t) be a smooth complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kähler. We prove that if ∣Rm(g(t))∣g(t) is bounded by a/t for some a > 0, then g(t) is Kähler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a smooth complete solution to the Kähler-Ricci flow on a noncompact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and ∣Rm(g(t))∣g(t) ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact Kähler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to ℂn; and (ii) there is ∈(n) > 0 depending only on n such that if (Mn,g0) is a complete noncompact Kähler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1+∈(n))-1h≤g0 ≤(1+∈(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to ℂn. [ABSTRACT FROM AUTHOR]

This paper considers the three-dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an L² maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the available estimates to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in spaces with critical regularity, giving solutions with bounded slope and time integrable bounded curvature. [ABSTRACT FROM AUTHOR]

In this paper we generalize our previous Local Removable Singularity Theorem for minimal laminations to the case of weak H-laminations (with H ∊ R constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mean curvature from leaf to leaf) of a compact Riemannian three-manifold N with boundary solely in terms of a bound of the absolute sectional curvature of N and of the distance to the boundary of N. We then apply these results to classify weak CMC foliations of R³ and S³ with a closed countable set of singularities. [ABSTRACT FROM AUTHOR]

We show that the scalar curvature is uniformly bounded for the normalized Kähler-Ricci flow on a Kähler manifold with semi-ample canonical bundle. In particular, the normalized Kähler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kähler surfaces, projective manifolds of complex dimension three, and for projective manifolds of all dimensions if assuming the abundance conjecture. [ABSTRACT FROM AUTHOR]

We give a condition under which the findings of the paper by Kenmotsu and Zhou [Amer . J. Math. 122 (2000), no. 2, 295-317] work well and determine the surfaces that were not considered before. [ABSTRACT FROM AUTHOR]

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples. [ABSTRACT FROM AUTHOR]

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in ℝn with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minimal surface or an inverted Chen's minimal graph must be close to these surfaces in the W2,2-norm. Moreover, we apply these results to prove a corresponding rigidity result for complete, connected and non-compact surfaces. [ABSTRACT FROM AUTHOR]

Let (M,g(t)) be a Kähler Ricci flow with positive first Chern class. First, we prove a uniform isoperimetric inequality for all time. In the process, we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on (M,g(t)) without assuming the Ricci curvature is bounded from below. [ABSTRACT FROM AUTHOR]

Based on work of Daniel and Mira, we prove that for each 0, there exists a unique H > 0, there exists a unique (up to ambient isometry) immersed constant mean curvature H sphere SH in Sol3, where Sol3 is the non-abelian solvable 3-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries; see Bonahon or Scott for a discussion of the Thurston geometries that can occur for compact 3-manifolds. Since this proof also demonstrates that SH has index one, the results in Daniel-Mira imply SH is embedded and the left-invariant Gauss map of SH is a diffeomorphism. One key new result that we prove in order to obtain these results for constant mean curvature spheres is the existence of height estimates for certain constant mean curvature H graphs in Sol3, which depend only on any fixed positive lower bound for H. By the work in Daniel-Mira, the existence of the spheres SH for all H > 0 implies the existence for all H ≥ 0 of a complex (holomorphic only when H = 0) quadratic differential QH defined on all constant mean curvature H surfaces in Sol3 and which, for H > 0, vanishes on an open set of M if and only if M is contained in some left translation of SH; furthermore, when QH is not identically zero on M, it has isolated zeros on and negative index at these zeros. [ABSTRACT FROM AUTHOR]

In this paper, we prove a general halfspace theorem for constant mean curvature surfaces. Under certain hypotheses, we prove that, in an ambient space M3³, any constant mean curvature H0 surface on one side of a constant mean curvature H0 surface Ε0 is an equidistant surface to Ε0. The main hypotheses of the theorem are that Ε0 is parabolic and the mean curvature of the equidistant surfaces to Ε0 evolves in a certain way. [ABSTRACT FROM AUTHOR]

Based on work of Daniel and Mira, we prove that for each 0, there exists a unique H > 0, there exists a unique (up to ambient isometry) immersed constant mean curvature H sphere SH in Sol3, where Sol3 is the non-abelian solvable 3-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries; see Bonahon or Scott for a discussion of the Thurston geometries that can occur for compact 3-manifolds. Since this proof also demonstrates that SH has index one, the results in Daniel-Mira imply SH is embedded and the left-invariant Gauss map of SH is a diffeomorphism. One key new result that we prove in order to obtain these results for constant mean curvature spheres is the existence of height estimates for certain constant mean curvature H graphs in Sol3, which depend only on any fixed positive lower bound for H. By the work in Daniel-Mira, the existence of the spheres SH for all H > 0 implies the existence for all H ≥ 0 of a complex (holomorphic only when H = 0) quadratic differential QH defined on all constant mean curvature H surfaces in Sol3 and which, for H > 0, vanishes on an open set of M if and only if M is contained in some left translation of SH; furthermore, when QH is not identically zero on M, it has isolated zeros on and negative index at these zeros. [ABSTRACT FROM AUTHOR]

We study the topology of the space ∂Kn of complete convex hypersurfaces of Rn which are homeomorphic to Rn-1. In particular, using Minkowski sums, we construct a deformation retraction of ∂Kn onto the Grassmannian space of hyperplanes. So every hypersurface in ∂K may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of ∂Kn consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg. [ABSTRACT FROM AUTHOR]

We prove that for every smooth compact Riemannian three-manifold ̅W with nonempty boundary, there exists a smooth properly embedded one-manifold Δ ⊂ W = Int(̅W), each of whose components is a simple closed curve and such that the domain D = W - Δ does not admit any properly immersed open surfaces with at least one annular end, bounded mean curvature, compact boundary (possibly empty) and a complete induced Riemannian metric. [ABSTRACT FROM AUTHOR]

We prove that a Riemannian manifold (M,g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains--so M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport. [ABSTRACT FROM AUTHOR]

All continuous linearly intertwining symmetric Blaschke valuations on convex bodies are completely classified. It is shown that there is a unique non-trivial such valuation. On symmetric bodies, this valuation is the curvature image operator. [ABSTRACT FROM AUTHOR]

In this paper we prove a maximum principle at infinity for properly embedded surfaces with constant mean curvature H > 0 in the 3-dimensional Euclidean space. We show that no one of these surfaces can lie in the mean convex side of another properly embedded H surface. We also prove that, under natural assumptions, if the surface lies in the slab [x3] < 1/2H and is symmetric with respect to the plane x3 = 0, then it intersects this plane in a countable union of strictly convex closed curves. [ABSTRACT FROM AUTHOR]

The Willmore energy of a surface immersed into ĝn is the integral of its squared mean curvature. It is known that the infimum of the Willmore energy among all closed oriented surfaces of genus p in ĝn is attained by a smooth embedded surface, and that 4π < βpn < 8 π for p ⩾ 1. We show that βpn converges to 8π as p goes to infinity. [ABSTRACT FROM AUTHOR]

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold N without supposing any sign condition on the sectional curvatures KN. A curvature flow wouldn't work in this Situation, neither the method of successive approximation. [ABSTRACT FROM AUTHOR]

One of the main purposes of this paper is to prove that on a complete Kahler manifold of dimension m, if the holomorphic bisectional curvature is bounded from below by -l and the minimum spectrum λ1(M) ⩾ m[sup2], then it must either be connected at infinity or isometric to R x N with a specialized metric, with N being compact. Generalizations to complete Kahler manifolds satisfying a weighted Poincaré inequality are also being considered [ABSTRACT FROM AUTHOR]

We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and construct gradient flows of functions on such spaces. If the underlying space is a Riemannian manifold of nonnegative sectional curvature, then our gradient flow of the free energy produces a solution of the linear Fokker-Planck equation. [ABSTRACT FROM AUTHOR]

Abstract. We extend the classical exsitence and uniqueness theory of Jenkins-Serrin (H = 0) and Spruck (H > 0) for the constant mean curvature equation over a domain in R[sub2], to domains in H[sub2] or S[sub2]. This theory allows prescribed boundary data including plus or minus infinity on entire arcs of the boundary. Necessarily these arc must have curvature +2H or -2H with respect to the domain. We give necessary and sufficient conditions for existence in terms of so called admissible polygons. The key idea, as in previous proofs, is to study the "flux" of monotone increasing and decreasing sequences of solutions. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we study the Calabi flow on a polarized Kähler manifold and some related problec~s. We first give a precise statement on the short time existence of the Calabi flow for any c3~(M) initial Kähler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Ktihler (CscK) metric will flow to a nearby CscK metric exponentially fast. Second~y, we prove that a compactness theorem in the space of the Ktihler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc. [ABSTRACT FROM AUTHOR]

We introduce a hyperbolic Gauss map into the Poincaré disk for any surface in H² × 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 can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H = 1/2 in H² × R. A similar result applies to minimal surfaces in the Heisenberg group Nil3. Finally, we classify all complete minimal vertical graphs in H² x IR. [ABSTRACT FROM AUTHOR]