Graph Translation Surface in the Lorentz-Heisenberg 3-space with constant curvatures

Yıl 2022, Cilt: 19 Sayı: 1, 13 - 21, 01.05.2022

Zoubır Hanifi , Brahim Medjahdı

Öz

In this paper, we study graph translation surfaces in a 3-dimensional Lorentz-Heisenberg
3-space H3. The classification theorems of the considered surfaces with zero and
nonzero mean and Gaussian curvatures are given. Contrary to the Euclidean case, there
is evidence that , translation surfaces with constant Gaussian curvature K that are not
cylindrical surfaces, with constant mean curvature H ̸= 0 which are not settled.

Anahtar Kelimeler

Lorentz-Heisenberg 3-space, Graph Translation Surface, Mean curvature, Gaussian curvature.

Kaynakça

• A. D. Polyanin, W. E. Schiesser and A. I. Zhurov, ”Partial differential equation,” Scholarpedia, vol. 3, no. 10, 2008.
• V. Ushakov, ”The explicit general solution of trivial Monge-Ampere equation,” Comment. Math. Helv., vol. 75, no. 1, pp. 125-133, 2000.
• R. Lopez, ”Separtion of variables in equations of mean-curvature type,” Proc. Roy. Soc. Edinburgh Sect., vol. 146, no 5, pp. 1017-1035, 2016.
• L. Simon, ”The minimal surface equation,” Geometry, Encyclopaedia Math. Sci. Springer Berlin, vol. 9, pp. 239-272,1997.
• L. Simon, ”Equations of mean curvature type in 2 independent variables,” Pacific j. Math, vol.69, no. 1, pp. 245-268, 1977.
• H. F. Sherk, ”Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen,” J. Rein. Angew. Math., vol. 13, pp. 185-208, 1835.
• H. Liu, ”Translation surfaces with constant mean curvature in three dimensional spaces,” Journal Geom., vol. 64, pp. 141-149, 1999.
• L. Verstraelen, J.Walrave and S. Yaprak, ”The minimalt ranslation surfaces in Euclidian space,” Soochow J. Math., vol.20, pp. 77-82, 1994.
• J. Inoguchi, R. Lopez and M. I. Munteanu, ”Minimal translation surfaces in the Heisenberg groub Nil3,” Geom. Dedicata., vol. 161, pp. 221-231, 2012.
• R. Lopez and M. I. Munteanu, ”Minimal translation surfaces in Sol3,” J. Math. Soc. Japan., vol. 64, pp. 985-1003, 2012.
• Dj. Bensikaddour and L. Belarbi, ”Minimal translation surfaces in Lorentz Heisenberg 3-space,” Nolinear Studies, vol.24, no. 4, pp. 859-867, 2017.
• C. Baba-Hamed, M. Bekkar and H. Zoubir, ”Translation surfaces of revolution in the 3-Dimensional LorentzMinkowski space satisfying ∆ri = λiri,” Int. Journal of Math. Analysis, vol. 4, pp. 797-808, 2010.
• D. W. Yoon, ”On the Gauss map of translation surfaces in Minkowski 3-space,” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 389-398, 2002.
• M. Munteanu, and A. I. Nistor, ”On the geometry of the second fundamental form of translation surfaces in E 3,” arXiv:0812.3166 [math.DG] 2008.
• A. Azzi, H. Zoubir and M. Bekkar, ”Surfaces of finite type in SL(2,R),” Nonlinear Studies, vol. 27, no. 3, pp. 783-794, 2020.
• N. Rahmani and S. Rahmani, ”Lorentzian geometry of the Heisenberg group,” Geom. Dedicata, vol. 118, pp. 133-140, 2006.
• N. Rahmani and S. Rahmani, ”Structures Homognes Lorentziennes sur le Groupe de Heisenberg,” J. Geom.Phys, vol. 13, pp. 254-258, 1994.