Timelike surfaces with zero mean curvature in Minkowski 4-space

TL;DR

This paper studies timelike surfaces with zero mean curvature in Minkowski 4-space, introducing canonical parameters, deriving PDE systems for their curvatures, and classifying rotational examples.

Contribution

It establishes a correspondence between solutions of PDE systems and timelike zero mean curvature surfaces, and classifies rotational examples in this class.

Findings

Gauss and normal curvatures satisfy a PDE system.

Solutions to the PDE system uniquely determine surfaces.

Rotational surfaces of Moore type are fully classified.

Abstract

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter family of solutions to the system of natural partial differential equations describing timelike surfaces with zero mean curvature.