Cycloidal maximal surfaces in the 3-dimensional Lorentz-Minkowski space

Abstract

We introduce new examples of maximal surfaces in Lorentz-Minkowski 3-dimensional space by solving the Bj\"orling problem for cycloidal curves and vector fields determined by their normal and the binormal vector fields.
We also provide their Weierstrass representation and determine the associated family of such surfaces. We analyze their geometric properties and show that one family of parametric curves of surfaces from associated family are generalized helices lying on a non-degenerated quadric.
Finally, we study a Lorentzian counterpart of the Henneberg surface and show that its adjoint is a maximal surface over a spacelike Lorentzian astroid.