On linear Weingarten surfaces

Details On linear Weingarten surfaces On linear Weingarten surfaces

2006-07-28

arxiv.org/abs/math/0607748v1

to appear in the International Journal of Mathematics, 2007

Mathematics

Differential Geometry

10 pages

Scientific paper

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.

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