Related papers: Properly embedded, area-minimizing surfaces in hyp…

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…

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…

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding…

We show that all hyperbolic surfaces admit an ideal triangulation with bounded shear parameters. This upper bound depends logarithmically on the topology of the surface.

We prove the existence of complete, embedded, constant mean curvature 1 surfaces in 3 dimensional hyperbolic space when g, the genus of the surface, and n, the number of ends of the surface, satisfy either g=0 and $n\geq 1$ or $g \geq 1$…

We consider properly immersed finite topology minimal surfaces S in complete finite volume hyperbolic 3-manifolds N, and in M x S(1), where M is a complete hyperbolic surface of finite area. We prove S has finite total curvature equal to…

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations…

We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.

In 1970, Lawson solved the topological realization problem for minimal surfaces in the sphere, showing that any closed orientable surface can be minimally embedded in $\mathbb{S}^3$. The analogous problem for surfaces with boundary was…

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…

We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one natural compactification of this space. This provides a…

On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies…

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by…

In this paper we consider three dimensional upper half space $\mathbb{H}^3 $ equipped with various Kropina metrics obtained by deformation of hyperbolic metric of $\mathbb{H}^3$ through $1$-forms and obtain a partial differential equation…

Objects with large symmetry groups have been an interest for many mathematicians. A classical question in geometry is whether a surface with certain geometric features, such as completeness, curvature, etc..., can embed in $\mathbb{R}^3.$…

The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex…

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and…

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal…

We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.