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

TL;DR

This paper classifies spacelike surfaces in Minkowski space satisfying a linear relation between principal curvatures, showing they are either rotational or Riemann examples of maximal surfaces, with explicit descriptions for certain cases.

Contribution

It provides a classification of linear Weingarten spacelike surfaces in Minkowski space, identifying conditions under which they are rotational or Riemann examples, and offers explicit solutions for specific cases.

Findings

Surfaces are either rotational or Riemann examples of maximal surfaces.

Explicit solutions are obtained for rotational surfaces with different axis types.

Complete descriptions are provided for the case when n=0.

Abstract

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.