Parabolic Weingarten surfaces in hyperbolic space

TL;DR

This paper classifies all parabolic surfaces in hyperbolic space that satisfy a linear relation between their principal curvatures, mean curvature, or Gaussian curvature, expanding understanding of their geometric properties.

Contribution

It provides a complete classification of parabolic Weingarten surfaces in hyperbolic space satisfying linear curvature relations, a previously unresolved problem.

Findings

Classification of all parabolic linear Weingarten surfaces in hyperbolic space

Explicit descriptions of these surfaces based on curvature relations

Extension of known results in hyperbolic geometry and surface theory

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.