Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$      

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Sufficient conditions for Willmore immersions in \mathbb{R }^3 to be minimal surfaces

We provide two sharp sufficient conditions for immersed Willmore surfaces in $\mathbb{R }^3$ to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with vanishing mean curvature on the boundary $\partial \varOmega $ must already be minimal graphs, which in particular yields some Bernstein-type result for Willmore graphs on $\mathbb{R }^2$ . Our methods also prove the non-existence of Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with mean curvature $H$ satisfying $H \ge c_0>0 \,{\text{ on }}\, \partial \varOmega $ if $\varOmega $ contains some closed disc of radius $\frac{1}{c_0} \in (0,\infty )$ , and they yield that any closed Willmore surface in $\mathbb{R }^3$ which can be represented as a smooth graph over $\mathbb{S }^2$ has to be a round sphere. Finally, we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford torus in $\mathbb{R }^3$ .     

Deformations of fundamental group representations and earthquakes on $${\textit{SO}}(n,1)$$ surface groups

In this article we construct a type of deformations of representations \(\pi _1(M)\rightarrow G\) where G is an arbitrary lie group and M is a large class of manifolds including \(\hbox {CAT}(0)\) manifolds. The deformations are defined based on codimension 1 hypersurfaces with certain conditions, and also on disjoint union of such hypersurfaces, i.e. multi-hypersurfaces. We show commutativity of deforming along disjoint hypersurfaces. As application, we consider Anosov surface groups in \({\textit{SO}}(n,1)\) and show that the construction can be extended continuously to measured laminations, thus obtaining earthquake deformations on these surface groups.     

Volumes for $${\mathrm{SL}}_N({\mathbb {R}})$$ , the Selberg Integral and Random Lattices

Foundations of Computational Mathematics - There is a natural left and right invariant Haar measure associated with the matrix groups GL $${}_N(\mathbb {R})$$ and SL $${}_N(\mathbb {R})$$ due to...     

Minimal Lagrangian surfaces in {\mathbb {CH}^2}and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of ?i?eica (or Toda) type to construct a minimal Lagrangian surface in ${\mathbb {CH}^2}$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the ${\mathbb {R}}$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the ?i?eica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.     

A classification of taut,Stein surfaces with a proper $${\mathbb{R}}$$-action

We present a classification of 2-dimensional, taut, Stein manifolds with a proper \mathbbR{\mathbb{R}}-action. For such manifolds the globalization with respect to the induced local \mathbbC{\mathbb{C}}-action turns out to be Stein. As an application we determine all 2-dimensional taut, non-complete, Hartogs domains over a Riemann surface.     

An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in . As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ_∞ is a Jordan curve homologous to zero in such that Γ_∞ is contained in a slab between two horizontal circles of with width equal to π. We construct vertical minimal graphs in over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in 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. The first author wish to thank Laboratoire Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu for the kind hospitality and support. The authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for partial financial support.     

Uniqueness of minimal surfaces whose boundary is a horizontal graph and some Bernstein problems in {{\mathbb{H}^{2}\times \mathbb{R}}}

We deduce that a connected compact immersed minimal surface in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ whose boundary has an injective horizontal projection on an admissible convex curve in ${\partial_\infty{\mathbb{H}^{2}\times \mathbb{R}}}$ , and satisfies an admissible bounded slope condition, is the Morrey’s solution of the Plateau problem and is a horizontal minimal graph. We prove that there is no entire horizontal minimal graph in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ .     

Height estimates for constant mean curvature graphs in $$\mathrm {Nil}_3$$Nil3 and $${\widetilde{PSL_2}}({\mathbb {R}})$$PSL2~(R)

Archiv der Mathematik - In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $$\mathrm {Nil}_3$$ and...     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Decompositions of {\mathbb {R}^n, n \ge 4} , into Convex Sets Generate Codimension 1 Manifold Factors

We show that if G is an upper semicontinuous decomposition ${\mathbb {R}^n, n \ge 4}$ , into convex sets, then the quotient space ${\mathbb {R}^n/G}$ is a codimension 1 manifold factor. In particular, we show that ${\mathbb {R}^n/G}$ has the disjoint arc-disk property.     

On differentiable compactifications of the hyperbolic plane and algebraic actions of $${\rm SL}_2(\mathbb{R})$$ on surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2;R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. The online version of the original article can be found under doi: .