Framed null curves and timelike surfaces via Lorentzian harmonic maps into de-Sitter 2-space

TL;DR

This paper constructs Lorentzian harmonic maps into de-Sitter space from framed null curves and explores associated timelike surfaces with constant mean curvature and minimal surfaces in Lorentzian 3-manifolds, analyzing their singularities.

Contribution

It introduces a new method to generate Lorentzian harmonic maps from framed null curves and studies their related timelike surfaces, including singularity characterization.

Findings

Constructed Lorentzian harmonic maps satisfying eigenvalue equations.

Characterized properties of singularities on timelike minimal surfaces.

Connected null curves with timelike surfaces in Lorentzian geometry.

Abstract

We construct a class of Lorentzian harmonic maps into the de-Sitter $2$-space satisfying the eigenvalue equation $\Box N=2H^2N$ for the d'Alambert operator $\Box$ and a non-zero constant $H$ from framed null curves. We also investigate two classes of timelike surfaces associated with these Lorentzian harmonic maps: the first one is timelike surfaces with constant mean curvature $H$ in Lorentz-Minkowski $3$-space and the second one is timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group $\operatorname{Nil}_3(H)$. In particular, we characterize some properties of singularities on timelike minimal surfaces in $\operatorname{Nil}_3(H)$ via an invariant of framed null curves.