Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$

Abstract

A maximal surface \({\mathcal{S}}\) with isolated singularities in a complete flat Lorentzian 3-manifold $$\mathcal{N}$$is said to be entire if it lifts to a (periodic) entire multigraph \({\sim\mathcal{S}}\) in \({\mathbb{L}^3}\) . In addition, \({\mathcal{S}}\) is called of finite type if it has finite topology, finitely many singular points and \({\sim\mathcal{S}}\) is a finitely sheeted multigraph. Complete (or proper) maximal immersions with isolated singularities in \({\mathcal{N}}\) are entire, and entire embedded maximal surfaces in \({\mathcal{N}}\) with a finite number of singularities are of finite type. We classify complete flat Lorentzian 3-manifolds carrying entire maximal surfaces of finite type, and deal with the topology, Weierstrass representation and asymptotic behavior of this kind of surfaces. Finally, we construct new examples of periodic entire embedded maximal surfaces in \({\mathbb{L}^3}\) with fundamental piece having finitely many singularities.