On linear Weingarten surfaces

TL;DR

This paper classifies linear Weingarten surfaces in Euclidean 3-space, showing that besides surfaces of revolution and Riemann's minimal surfaces, no other such surfaces parametrized by a family of circles exist.

Contribution

It provides a classification of linear Weingarten surfaces parametrized by circles, identifying the unique cases of revolution and Riemann's minimal surfaces.

Findings

Surfaces of revolution satisfy the linear Weingarten condition.

Riemann's minimal surfaces are the only non-revolution examples.

No other circle-parametrized linear Weingarten surfaces exist besides these cases.

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.