On linear Weingarten surfaces

Abstract

In this paper we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as ${\kappa }_1=m{\kappa }_2+n$, where $m$ and $n$ are real numbers and ${\kappa }_1$ and ${\kappa }_2$ denote the principal curvatures at each point of the surface. We investigate the possible existence of such surfaces parametrized by a uniparametric family of circles. Besides the surfaces of revolution, we prove that not exist more except the case $(m,n)=(-1,0)$, that is, if the surface is one of the classical examples of minimal surfaces discovered by Riemann.

10 pages