Monatsh. Math. 140, 315–334 (2003) DOI 10.1007/s00605-003-0111-9 Helicoidal Maximal Surfaces in Lorentz-Minkowski Space By Pablo Mira 1 and Jose ´ A. Pastor 2 1 Universidad Polit ecnica de Cartagena, Cartagena, Spain 2 Universidad de Murcia, Espinardo, Spain Received June 19, 2002; in revised form February 4, 2003 Published online November 28, 2003 # Springer-Verlag 2003 Abstract. We investigate helicoidal surfaces in Lorentz-Minkowski space L 3 . A purely geometric method is used for classifying the helicoidal maximal surfaces. This description consists on exposing helicoidal maximal surfaces as solutions to certain adequate Bj€ orling problems in L 3 , and shows that a maximal surface is helicoidal if and only if it lies in the associate family of a catenoid in L 3 . We shall also give characteristic properties of the catenoids and helicoids in L 3 . 2000 Mathematics Subject Classification: 53A10, 53C50 Key words: Minkowski space, helicoidal surface, Bj€ orling problem, maximal surface, associate family 1. Introduction One of the most interesting features of the local surface theory in Lorentz- Minkowski space L 3 is that, even though it behaves quite like the classical surface theory in Euclidean space, there are sometimes striking differences. Perhaps the most representative one is the existence of a certain type of Lorentzian surfaces, called flat B-scrolls, discovered by Graves [11]. These surfaces are isometric immer- sions from L 2 to L 3 with non-diagonalizable shape operator. They are flat and mini- mal but not totally geodesic, and have appeared completing many classification results for surfaces in L 3 [2, 8, 10, 11, 26]. More recently, Dillen and K€ uhnel [9] investigated helicoidal motions in L 3 and exhibited a special type of them that does not have an Euclidean counterpart. They christened them as cubic screw motions. Motivated by [9], the present paper studies helicoidal surfaces in L 3 , that is, semi-Riemannian surfaces in L 3 that are invariant under a helicoidal motion group. We will relate flat B-scrolls with cubic screw motions. Actually we shall construct a special type of B-scroll, which we call parabolic null cylinder, that is invariant under a group of cubic screw motions. Even though from the Euclidean viewpoint this is nothing but a right cylinder over a parabola, it has interesting properties when seen as a helicoidal surface in L 3 . For instance, whereas a right circular cylinder in L 3 is helicoidal for a specific axis, and for every pitch, these parabolic null cylinders are helicoidal just for one pitch, but for infinitely many axes. In particular, they are not rotation surfaces in L 3 .