Related papers: Asymptotic H-Plateau Problem in Hyperbolic 3-space
In this paper are studied the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of…
In this paper, we show that any open orientable surface S can be properly embedded in H^3 as a minimizing H-surface for any 0<=H<1. We obtained this result by proving a version of the bridge principle at infinity for H-surfaces. We also…
In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a…
Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $\epsilon\approx 0$, a closed and embedded curve $u^\epsilon $ in $\mathbb H^2$ having geodesic curvature…
A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…
We prove that given a finite set $E$ in a bordered Riemann surface $\mathcal{R}$, there is a continuous map $h\colon \overline{\mathcal{R}}\setminus E\to\mathbb{C}^n$ ($n\geq 2$) such that $h|_{\mathcal{R}\setminus E} \colon…
We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a…
We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the…
We present a proof that the hyperbolic plane cannot be isometrically immersed in Euclidean $3$-space by a $C^\infty$ map. Ideas from many topics in (essentially) undergraduate mathematics are applied; the use of moving frames and connection…
In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones…
We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…
We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\"uller space of S, with quasi-convex image lying in the thick part. As a consequence,…
Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed…
We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for…
In [14], it was shown that, if M is a 3-dimensional asymptotically harmonic with minimal horospheres, then M is flat. However, there is a gap in the proof of this paper. In this paper, we provide the correct proof of the result. Thus we…
We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.
We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends…
In this paper, we show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic $3$-manifolds except some special cases.
We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space $\mathbb{H}^3$. We go further and establish a jet interpolation theorem…