Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 219-230.

NULL-TRANSLATION SURFACES WITH CONSTANT CURVATURES IN LORENTZ-MINKOWSKI 3-SPACE

Ivana Filipan, Željka Milin Šipuš and Ljiljana Primorac Gajčić

Faculty of Science, Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: milin@math.hr

Department of Mathematics, University J. J. Strossmayer of Osijek, 31 000 Osijek, Croatia
e-mail: ljiljana.primorac@mathos.hr

Abstract.   Translation surface is a surface formed by two curves moving along each other. In this paper we analyze this kind of surfaces in Lorentz-Minkowski 3-space R₁³, which is the smooth manifold R³ endowed by a flat Lorentzian pseudometric. Translation surfaces in R₁³ can be classified with respect to the causal character of their generating curves (spacelike, timelike or null (lightlike)). We are specially interested in translation surfaces generated by at least one null curve, which we refer to as null-translation surfaces. In the present paper we determine all nulltranslation surfaces of constant mean curvature and prove that the only null-translation surfaces of constant Gaussian curvature are cylindrical surfaces.

2020 Mathematics Subject Classification.   53A10, 53A35, 53A55.

Key words and phrases.   Translation surface, null curve, mean curvature, Gaussian curvature.

DOI: https://doi.org/10.21857/yk3jwhnl19

References:

1. F. Dillen, W. Goemans and I. Van De Woestyne, Translation surfaces of Weingarten type in 3-space, Bull. Transilv. Univ. Brasov Ser. III, Math. Inform. Phys. 1(50) (2008), 109-122.
   MathSciNet

2. V. P. Gorokh, Two-dimensional minimal surfaces in a pseudo-Euclidean space, J. Soviet Math. 54 (1991), 691-702.
   MathSciNet     CrossRef

3. T. Hasanis and R. López, Translation surfaces in Euclidean space with constant Gaussian curvature, Comm. Anal. Geom. 6(29) (2021), 1415-1447.
   MathSciNet     CrossRef

4. J. I. Inoguchi and S. Lee, Null curves in Minkowski 3-space, Int. Electron. J. Geom. 1 (2008), no.2, 40-83.
   MathSciNet

5. S. Lee and J. H. Varnado, Timelike surfaces of revolution with constant mean curvature in Minkowski 3-space, Differ. Geom. Dyn. Syst. 7 (2007) 82-102.
   MathSciNet

6. H. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999) 141-149.
   MathSciNet     CrossRef

7. R. López, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom. 7 (2014), 44-107.
   MathSciNet

8. R. López and M. Moruz, Translation and homothetical surfaces in Euclidean space with constant curvature, J. Korean Math. Soc. 52 (2015), 523-535.
   MathSciNet     CrossRef

9. R. López and Ó. Perdomo, Minimal translation surfaces in Euclidean space, J. Geom. Anal. 27 (2017), 2926-2937.
   MathSciNet     CrossRef

10. R. López and E. Demir Cetin, Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, Centr. Eur. J. Math. 12 (2014), 1349-1361.
    MathSciNet     CrossRef

11. F. Manhart, Timelike helicoidal surfaces in Lorentz-Minkowski 3-space revisited, Slovak J. Geom. Graph. 12(24) (2015), 27-43.

12. L. V. McNertney, One-parameter families of surfaces with constant curvature in Lorentz 3-space, PhD Thesis, Brown University, 1980.

13. Ž. Milin Šipuš, Translation surfaces of constant curvature in a simply isotropic space, Period. Math. Hung. 68 (2014), 160-175.
    MathSciNet     CrossRef

14. Ž. Milin Šipuš and B. Divjak, Translation surfaces in the Galilean space, Glas. Mat. Ser. III 46(66) (2011), 455-469.
    MathSciNet     CrossRef