Related papers: Asymptotic H-Plateau Problem in Hyperbolic 3-space

We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the…

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a \pi_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has…

We prove that a finite type curve is an $\xi$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $\xi$-asymptotic line. These results obtained here…

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover,…

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.

In this paper, we express surfaces parametrically through a given spacelike (timelike) asymptotic curve using the Frenet frame of the curve in Minkowski 3-space. Necessary and sufficient conditions for the coefficients of the Frenet frame…

We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient…

We develop a min-max theory for certain complete minimal hypersurfaces in hyperbolic space. In particular, we show that given two strictly stable minimal hypersurfaces that are both asymptotic to the same ideal boundary, there is a new one…

We construct new examples of Kobayashi hyperbolic hypersurfaces in the projective 4-space. They are generic projections of the triple symmetric product of a generic curve of genus at least 7, smoothly embedded in the projective 7-space.

We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit configuration spaces, differential and…

We prove that given two compact oriented $3$-manifolds $N$ and $M,$ with $M$ satisfying only a mild hypothesis, there is a hyperbolic $3$-manifold $N'$ arbitrarily ``closely related'' to $N,$ and such that $N'$ does not embed in $M.$ For…

We construct simply connected, complete, non-$CMC$ biconservative surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set…

We study the asymptotic Plateau problem in $\mathbb{H}_2\times \mathbb{R}$. We give the first examples of non-fillable finite curves with no thin tail in the asymptotic cylinder. Furthermore, we study the fillability question for infinite…

In this paper, we improve a result by Chodosh and Ketover. We prove that, in an asymptotically flat $3$-manifold $M$ that contains no closed minimal surfaces, fixing $q\in M$ and a $2$-plane $V$ in $T_qM$ there is a properly embedded…

Given a constant $k>1$, let $Z$ be the family of round spheres of radius $\textrm{artanh}(k^{-1})$ in the hyperbolic space $\mathbb{H}^3$, so that any sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result involving…

We show a generic finiteness result for least area planes in 3-dimensional hyperbolic space. Moreover, we prove that the space of minimal immersions of disk into hyperbolic space is a submanifold of a product bundle over a space of…

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is…

This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,\om) when J\in \Jj(\Ss), the set of \om-tame J for which a fixed chain \Ss of transversally intersecting embedded spheres of self-intersection \le -2 is…

Let $\mathbb{H}^n$ be the $n-$dimensional hyperbolic space. It is well known that, if $f: \mathbb{H}^n\to \mathbb{H}^n$ is a bijection that preserves $r-$dimensional hyperplanes, then $f$ is an isometry. In this paper we make neither…