Parabolic Weingarten surfaces in hyperbolic space

Abstract

A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form $a{\kappa }_1+b{\kappa }_2=c$ or $aH+bK=c$, where $a,b,c\in \r$ and, as usual, ${\kappa }_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in hyperbolic space.

22 pages, 10 figures; This work was announced in arXiv:0704.2755