Spacelike surfaces in Minkowski 4-space with a canonical normal null direction

TL;DR

This paper investigates spacelike surfaces in Minkowski 4-space with a special null direction, characterizing their geometry, properties, and methods of construction, including a PDE approach.

Contribution

It provides a detailed description and characterization of spacelike surfaces with a canonical null normal direction, introducing new geometric insights and construction methods.

Findings

Characterization of surfaces with canonical null normal direction

Analysis of the Gauss map and curvature ellipse

Two methods for constructing such surfaces, including a PDE approach

Abstract

A canonical normal null direction on a spacelike surface in the four dimensional Minkowski space $\mathbb{R}^{3,1}$ is a parallel vector field $Z$ on $\mathbb{R}^{3,1}$ such that the normal component of $Z$ on the surface is a lightlike vector field. We describe the geometric properties of a spacelike surface endowed with a canonical normal null direction and we obtain some characterizations of these surfaces. Moreover, using their Gauss map we study other properties of these surface: the associated ellipse of curvature and their asymptotic directions. Finally, we give two different ways to create these surfaces, one of them involves a nonlinear partial differential equation.