Analysis of timelike Thomsen surfaces - with deformations and singularities

TL;DR

This paper provides a comprehensive classification of timelike Thomsen surfaces, a special class of timelike minimal surfaces, using conformal and null coordinates, and explores their deformations and singularities.

Contribution

It offers a complete global classification of timelike Thomsen surfaces and characterizes them via lightlike curvatures, revealing their relation to timelike minimal surfaces with planar curvature lines.

Findings

Complete classification of timelike Thomsen surfaces.

Characterization using lightlike curvatures.

Deformation of null curves preserving key properties.

Abstract

Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and to characterize them using a geometric invariant called lightlike curvatures. As a result, we reveal the relationship between timelike Thomsen surfaces, and timelike minimal surfaces with planar curvature lines. As an application, we give a deformation of null curves preserving the pseudo-arclength parametrization and the constancy of the lightlike curvatures.