Parabolic surfaces in hyperbolic space with constant curvature

TL;DR

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

Contribution

It provides a classification of parabolic Weingarten surfaces in hyperbolic space with constant curvature or linear curvature relations, which was previously unexplored.

Findings

Classification of parabolic surfaces with constant Gaussian curvature.

Classification of surfaces satisfying linear curvature relations.

Enhanced understanding of geometric properties of these surfaces.

Abstract

We study parabolic linear Weingarten surfaces in hyperbolic space $\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $a\kappa_1+b\kappa_2=c$, where $\kappa_i$ are the principal curvatures, and $a,b$ and $c$ are constant.