Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. I

A ug 2 00 0 MEAN CURVATURE 1 SURFACES IN HYPERBOLIC 3 - SPACE WITH LOW TOTAL CURVATURE I

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature −1 has two natural notions of “total curvature”— one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. In thi...

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...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Total Curvature of Complete Surfaces in Hyperbolic Space

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour.

Surfaces of Revolution with Constant Mean Curvature in Hyperbolic 3-Space

In this paper, we construct surfaces of revolution with constant mean curvature H = c and minimal surfaces of revolution in hyperbolic 3-space H(−c) of constant sectional curvature −c. It is shown that surfaces of revolution with constant mean curvature H = c in H(−c) tend toward the catenoid, the minimal surface of revolution in Euclidean 3-space E as c → 0. Minimal surfaces of revolution in H...