Spacelike surfaces in Minkowski space satisfying a linear relation between their principal curvatures

Details Spacelike surfaces in Minkowski space satisfying a linear relation between their principal curvatures Spacelike surfaces in Minkowski space satisfying a linear relation between their principal curvatures

2010-03-23

arxiv.org/abs/1003.4550v1

Mathematics

Differential Geometry

14 pages, 1 figure

Scientific paper

In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}%_{1}^{3}$ that satisfy a linear Weingarten condition of type $\kappa_{1}=m\kappa_{2}+n$, where $m$ and $n$ are constant and $\kappa_{1}$ and $\kappa_{2}$ denote the principal curvatures at each point of the surface. We study the family of surfaces foliated by a uniparametric family of circles in parallel planes. We prove that the surface must be rotational or the surface is part of the family of Riemann examples of maximal surfaces ($m=-1$, $n=0$). Finally, we consider the class of rotational surfaces for the case $n=0$, obtaining a first integration if the axis is timelike and spacelike and a complete description if the axis is lightlike.

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