Acknowledgement
The authors would like to thank the Referees for all helpful comments and suggestions that have improved the quality of our initial manuscript. The authors were supported by The National Agency Scientific Research (DGRSDT).
References
- U. Abresch and H. Rozenberg, A Hopf differential for constant mean curvature surfaces in S2 × ℝ and ℍ2 × ℝ, Acta Math. 193 (2004), 141-174. https://doi.org/10.1007/BF02392562
- L. Belarbi, On the symmetries of the Sol3 Lie group, J. Korean Math. Soc. 57 (2020), 523-537. https://doi.org/10.4134/JKMS.j190198
- L. Belarbi, Surfaces with constant extrinsically Guassian curvature in the Heisenberg group, Ann. Math. Inform. 50 (2019), 5-17.
- L. Belarbi and M. Belkhelfa, On The Ruled Minimal Surfaces in Heisenberg 3-Space With Density, Journal of Interdisciplinary Mathematics 23 (2020), 1141-1155. https://doi.org/10.1080/09720502.2020.1740498
- D. Bensikaddour and L. Belarbi, Minimal Translation Surfaces in Lorentz-Heiesenberg 3-Space, Nonlinear Stud. 24 (2017), 859-867.
- D. Bensikaddour and L. Belarbi, Minimal Translation Surfaces in Lorentz Heisenberg Space (H3, g2), Journal of Interdisciplinary Mathematics 24 (2021), 881-896. https://doi.org/10.1080/09720502.2020.1815345
- D. Bensikaddour and L. Belarbi, Minimal Translation Surfaces in Lorentz-Heiesenberg 3-space with Flat Metric, Differential Geometry-Dynamical Systems 20 (2018), 1-14.
- F. Bonahon, Geometric structures on 3-manifolds, In Handbook of geometric topology, North-Holland, Amsterdam, 2002, 93-164.
- R. Cadeo, P. Piu and A. Ratto, SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces, Maniscripta Math. 87 (1995), 1-12. https://doi.org/10.1007/BF02570457
- B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87-131. https://doi.org/10.4171/CMH/86
- R. Sa Erap and E. Toubiana, Screw motion surfaces in ℍ2 × ℝ and S2 × ℝ, Illinois J. Math. 49 (2005), 1323-1362. https://doi.org/10.1215/ijm/1258138140
- J. Inoguchi, Flat translation surfaces in the 3-dimensional Heisenberg group, J. Geom. 82 (2005), 83-90. https://doi.org/10.1007/s00022-005-1730-1
- R. Lopez, Constant mean curvature surfaces in Sol with non-empty boundary, Houston. J. Math. 38 (2012), 1091-1105.
- R. Lopez and M.I. Munteanu, Invariant surfaces in homogeneous space Sol with constant curvature, Math. Nach. 287 (2014), 1013-1024. https://doi.org/10.1002/mana.201010083
- R. Lopez and M.I. Munteanu, Minimal translation surfaces in Sol3, J. Math. Soc. Japan 64 (2012), 985-1003. https://doi.org/10.2969/jmsj/06430985
- J.M. Manzano and R. Souam, The classification of totally ombilical surfaces in homogeneous 3-manifolds, Math. Z. 279 (2015), 557-576. https://doi.org/10.1007/s00209-014-1381-8
- W.S. Massey, Surfaces of Gaussian curvature zero in euclidean 3-space, Tohoku Math. J. 14 (1962), 73-79. https://doi.org/10.2748/tmj/1178244205
- W.H. Meeks, Constant mean curvature spheres in Sol3, Amer. J. Math. 135 (2013), 1-13. https://doi.org/10.1353/ajm.2013.0011
- W.H. Meeks and J. Perez, Constant mean curvature in metric Lie groups, Contemp. Math. 570 (2012), 25-110. https://doi.org/10.1090/conm/570/11304
- W.H. Meeks III and H. Rosenberg, The theory of minimal surfaces in 𝕄 × ℝ, Comment. Math. Helv. 80 (2005), 811-885.
- B. Nelli and H. Rozenberg, Minimal surfaces in ℍ2 × ℝ, Bull. Braz. Math. Soc. 33 (2002), 263-292. https://doi.org/10.1007/s005740200013
- H. Rosenberg, Minimal surfaces in 𝕄2 × ℝ, Illinois J. Math. 46 (2002), 1177-1195. https://doi.org/10.1215/ijm/1258138473
- P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487. https://doi.org/10.1112/blms/15.5.401
- R. Souam and E. Toubiana, On the classification and regularity of umbilic surfaces in homogeneous 3-manifolds, Mat. Contemp. 30 (2006), 201-215.
- R. Souam and E. Toubiana, Totally umbilic surfaces in homogeneous 3-manifolds, Comm. Math. Helv. 84 (2009), 673-704.
- R. Souam, On stable constant mean curvature surfaces in S2 ×ℝ and ℍ2 ×ℝ, Trans. Amer. Math. Soc. 362 (2010), 2845-2857. https://doi.org/10.1090/S0002-9947-10-04826-9
- W.M. Thurston, Three-dimensional Geometry and Topology I, Princeton Math. Series(Levi, S. ed) 1997.