Maximal translation surfaces in Lorentz-Minkowski space

TL;DR

This paper classifies maximal translation surfaces in Lorentz-Minkowski space, showing planar curve cases, generalizing Scherk surfaces, and introducing new examples with pseudo-null curves.

Contribution

It provides a complete classification of maximal translation surfaces, including new examples involving pseudo-null curves, extending known Euclidean results.

Findings

Planar generating curves imply the other is planar.

Generalization of Scherk surfaces to Lorentz-Minkowski space.

New maximal surfaces with pseudo-null curves are constructed.

Abstract

A translation surface in Lorentz-Minkowski space $\rr^3$ is a surface defined as the sum of two spatial curves. In this paper we present a classification of maximal surfaces of translation type. We prove that if a generating curve is planar, then the other generating curve is also planar. We give a full description of these surfaces. In case that both curves are of Frenet type, we generalize the Scherk surfaces. In case that a curve is a pseudo-null curve, we obtain new examples of maximal surfaces which have not counterparts in Euclidean space.