Rotational linear Weingarten surfaces of hyperbolic type

Abstract

A linear Weingarten surface in Euclidean space ${\mathbf{R}}^3$ is a surface whose mean curvature $H$ and Gaussian curvature $K$ satisfy a relation of the form $aH+bK=c$, where $a,b,c\in \mathbf{R}$. Such a surface is said to be hyperbolic when $a^2+4bc<0$. In this paper we classify all rotational linear Weingarten surfaces of hyperbolic type. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ${\mathbf{R}}^3$ that consists into periodic surfaces with self-intersections.

15 pages, 4 figures