Weierstrass Representation of Lightlike Surfaces in Lorentz-Minkowski 4-Space

Year 2023, Volume: 16 Issue: 1, 232 - 243, 30.04.2023

Davor Devald Z. Milin Sipus

https://doi.org/10.36890/iejg.1272924

Abstract

We present a Weierstrass-type representation formula which locally represents every regular two-dimensional lightlike surface in Lorentz-Minkowski 4-Space $\mathbb{M}^4$ by three dual functions $(\rho,f,g)$ and generalizes the representation for regular lightlike surfaces in $\mathbb{M}^3$. We give necessary and sufficient conditions on the functions $\rho$, $f$, $g$ for the surface to be minimal, ruled or $l$-minimal. For ruled lightlike surfaces, we give necessary and sufficient conditions for the representation itself to be ruled. Furthermore, we give a result on totally geodesic half-lightlike surfaces which holds only in $\mathbb{M}^4$.

Keywords

Weierstrass representation, lightlike surface, minimal surface, conformal parametrization, ruled surface, Lorentz-Minkowski 4-Space

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