Ruled Laguerre minimal surfaces

A Laguerre minimal surface is an immersed surface in being an extremal of the functional . In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces , where are fixed. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.