Surfaces and Fronts with Harmonic-mean Curvature One in Hyperbolic Three-space

Hyperbolic surfaces of $L_1$-2-type

In this paper, we show that an $L_1$-2-type surface in the three-dimensional hyperbolic space $H^3subset R^4_1$ either is an open piece of a standard Riemannian product $ H^1(-sqrt{1+r^2})times S^{1}(r)$, or it has non constant mean curvature, non constant Gaussian curvature, and non constant principal curvatures.

On the Geometry of Constant Mean Curvature One Surfaces in Hyperbolic Space

We give a geometric classification of regular ends with constant mean curvature 1 and finite total curvature, embedded in hyperbolic space. We prove that each such end is either asymptotic to a catenoid cousin or asymptotic to a horosphere. We also study symmetry properties of constant mean curvature 1 surfaces in hyperbolic space associated to minimal surfaces in Euclidean space. We describe t...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Mean Curvature One Surfaces in Hyperbolic Space, and Their Relationship to Minimal Surfaces in Euclidean Space

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.

Constant Mean Curvature Surfaces with Two Ends in Hyperbolic Space

We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are well-understood surfaces of revolution – the catenoid cousins. In contrast to t...