Singular Miminal Ruled Surfaces

Abstract

In this paper we study surfaces with minimal potential energy under gravitational forces, called singular minimal surfaces. We prove that a singular minimal ruled surface in a Euclidean $3-$space is cylindrical, in particular as an $\alpha -$catenary cylinder by a result of López [Ann. Glob. Anal. Geom. 53(4) (2018), 521-541]. This result is also extended in Lorentz-Minkowski $3-$space.