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

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,

PDF

___

[1] Bejancu, A., Ferrandez, A., Lucas, P.: A new viewpoint on geometry of a lightlike hypersurface in semi-Euclidean space. Saitama Math. J. 16, 31-38 (1998).
[1] Bejancu, A., Ferrandez, A., Lucas, P.: A new viewpoint on geometry of a lightlike hypersurface in semi-Euclidean space. Saitama Math. J. 16, 31-38 (1998).
[2] Devald, D.: Weierstrass Representation for Timelike Surfaces in Minkowski 4-Space. J. Geom. 113 (2021). https://doi.org/10.1007/s00022-021- 00587-2
[2] Devald, D.: Weierstrass Representation for Timelike Surfaces in Minkowski 4-Space. J. Geom. 113 (2021). https://doi.org/10.1007/s00022-021- 00587-2
[3] Devald, D., Milin Šipuš, Ž.: Weierstrass Representation for Lightlike Surfaces in Lorentz-Minkowski 3-Space. J. Geom. Phys. 166 (2021). https://doi.org/10.1007/s00022-021-00587-2
[3] Devald, D., Milin Šipuš, Ž.: Weierstrass Representation for Lightlike Surfaces in Lorentz-Minkowski 3-Space. J. Geom. Phys. 166 (2021). https://doi.org/10.1007/s00022-021-00587-2
[4] do Carmo, M. P.: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., (1976).
[4] do Carmo, M. P.: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., (1976).
[5] Duggal, K. L., Bejancu, A.: Lightlike Submanifolds of semi-Riemannian Manifolds and Applications. Kluwer Academic Publishers, (1996).
[5] Duggal, K. L., Bejancu, A.: Lightlike Submanifolds of semi-Riemannian Manifolds and Applications. Kluwer Academic Publishers, (1996).
[6] Duggal, K. L., ¸Sahin, B.: Differential geometry of lightlike submanifolds. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2010).
[6] Duggal, K. L., ¸Sahin, B.: Differential geometry of lightlike submanifolds. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2010).
[7] Gorkaviy, V.: On Minimal Lightlike Surfaces in Minkowski Space-time. Differ. Geom. Appl. 26, 133-139 (2008). https://doi.org/10.1016/j.difgeo.2007.11.016
[7] Gorkaviy, V.: On Minimal Lightlike Surfaces in Minkowski Space-time. Differ. Geom. Appl. 26, 133-139 (2008). https://doi.org/10.1016/j.difgeo.2007.11.016
[8] Inoguchi, J., Lee, S.: Lightlike surfaces in Minkowski 3-Space. Int. J. Geom. Methods Mod. Phys. 6, 267-283 (2009). https://doi.org/10.1142/S0219887809003552
[8] Inoguchi, J., Lee, S.: Lightlike surfaces in Minkowski 3-Space. Int. J. Geom. Methods Mod. Phys. 6, 267-283 (2009). https://doi.org/10.1142/S0219887809003552
[9] Konopelchenko B. G.: Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy. Ann. Glob. Anal. Geom. 18, 67-74 (2000). https://doi.org/10.1023/A:1006608908156
[9] Konopelchenko B. G.: Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy. Ann. Glob. Anal. Geom. 18, 67-74 (2000). https://doi.org/10.1023/A:1006608908156
[10] Konopelchenko B. G., Landolfi G.: Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces. J. Geom. Phys. 29, 319-333 (1999). https://doi.org/10.1016/S0393-0440(98)00046-1
[10] Konopelchenko B. G., Landolfi G.: Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces. J. Geom. Phys. 29, 319-333 (1999). https://doi.org/10.1016/S0393-0440(98)00046-1
[11] Lee, S.: Weierstrass representation for timelike minimal surfaces in Minkowski 3-Space. Commun. Math. Anal., Conf 01, 11-19 (2008).
[11] Lee, S.: Weierstrass representation for timelike minimal surfaces in Minkowski 3-Space. Commun. Math. Anal., Conf 01, 11-19 (2008).
[12] Liu, H.: Weierstrass type representation for marginally trapped surfaces in Minkowski 4-Space. Math. Phys. Anal. Geom. 16, 171-178 (2013).
[12] Liu, H.: Weierstrass type representation for marginally trapped surfaces in Minkowski 4-Space. Math. Phys. Anal. Geom. 16, 171-178 (2013).
[13] Magid, M. A.: Timelike surfaces in Lorentz 3-Space with prescribed mean curvature and Gauss map. Hokkaido Math. J. 20, 447-464 (1991).
[13] Magid, M. A.: Timelike surfaces in Lorentz 3-Space with prescribed mean curvature and Gauss map. Hokkaido Math. J. 20, 447-464 (1991).
[14] McNertney, L. V.: One-parameter Families of Surfaces With Constant Curvature in Lorentz 3-Space. Ph.D. thesis. Brown University (1980).
[14] McNertney, L. V.: One-parameter Families of Surfaces With Constant Curvature in Lorentz 3-Space. Ph.D. thesis. Brown University (1980).
[15] Messelmi, F.: Analysis of Dual Functions. Annual Review of Chaos Theory, Bifurcations and Dynamical Systems. 4, 37-54 (2013
[15] Messelmi, F.: Analysis of Dual Functions. Annual Review of Chaos Theory, Bifurcations and Dynamical Systems. 4, 37-54 (2013