Home On minimal surfaces in ℍ2 × ℝ space
Article
Licensed
Unlicensed Requires Authentication

On minimal surfaces in ℍ2 × ℝ space

  • Bendehiba Senoussi EMAIL logo
Published/Copyright: August 3, 2024
Become an author with De Gruyter Brill

Abstract

A surface is minimal if the mean curvature mean vanishes everywhere. In this paper, we study some surfaces in the product space 2 × . In particular, we completely classify minimal surfaces.

Acknowledgements

The author wishes to express their sincere thanks to the referee for making several useful comments.

References

[1] U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in 𝐒 2 × 𝐑 and 𝐇 2 × 𝐑 , Acta Math. 193 (2004), no. 2, 141–174. 10.1007/BF02392562Search in Google Scholar

[2] M. Bekkar, Exemples de surfaces minimales dans l’espace de Heisenberg, Rend. Semin. Fac. Sci. Univ. Cagliari 61 (1991), no. 2, 123–130. Search in Google Scholar

[3] M. Bekkar, F. Bouziani, Y. Boukhatem and J. Inoguchi, Helicoids and axially symmetric minimal surfaces in 3-dimensional homogeneous spaces, Differ. Geom. Dyn. Syst. 9 (2007), 21–39. Search in Google Scholar

[4] J. A. Gálvez and J. L. Teruel, Complete surfaces with negative extrinsic curvature in 𝕄 2 × , J. Math. Anal. Appl. 423 (2015), no. 1, 538–546. 10.1016/j.jmaa.2014.10.002Search in Google Scholar

[5] J. Inoguchi, R. López and M.-I. Munteanu, Minimal translation surfaces in the Heisenberg group Nil 3 , Geom. Dedicata 161 (2012), 221–231. 10.1007/s10711-012-9702-8Search in Google Scholar

[6] R. López and M. I. Munteanu, Minimal translation surfaces in Sol 3 , J. Math. Soc. Japan 64 (2012), no. 3, 985–1003. 10.2969/jmsj/06430985Search in Google Scholar

[7] L. A. Masal’tsev, Minimal ruled surfaces in the three-dimensional geometries S 2 𝐑 and H 2 𝐑 (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (2004), no. 9, 46–52; translation in Russian Math. (Iz. VUZ) 48 (2004), no. 9, 43–49. Search in Google Scholar

[8] E. Molnár, The projective interpretation of the eight 3-dimensional homogeneous geometries, Beiträge Algebra Geom. 38 (1997), no. 2, 261–288. Search in Google Scholar

[9] S. Montaldo and I. I. Onnis, Invariant surfaces of a three-dimensional manifold with constant Gauss curvature, J. Geom. Phys. 55 (2005), no. 4, 440–449. 10.1016/j.geomphys.2005.01.002Search in Google Scholar

[10] S. Montaldo and I. I. Onnis, A note on surfaces in 2 × , Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), no. 3, 939–950. Search in Google Scholar

[11] E. Molnár and J. Szirmai, Symmetries in the 8 homogeneous 3-geometries, Symmetry Cult. Sci. 21 (2010), no. 1–3, 87–117. Search in Google Scholar

[12] B. Nelli and H. Rosenberg, Minimal surfaces in 2 × , Bull. Braz. Math. Soc. (N. S.) 33 (2002), no. 2, 263–292. 10.1007/s005740200013Search in Google Scholar

[13] H. Rosenberg, Minimal surfaces in 𝕄 2 × , Illinois J. Math. 46 (2002), no. 4, 1177–1195. 10.1215/ijm/1258138473Search in Google Scholar

[14] R. Sa Earp and E. Toubiana, Screw motion surfaces in 2 × and 𝕊 2 × , Illinois J. Math. 49 (2005), no. 4, 1323–1362. Search in Google Scholar

[15] D. W. Yoon, Some surfaces with zero curvature in 2 × , J. Appl. Math. (2014), Article ID 154294. Search in Google Scholar

Received: 2023-10-12
Revised: 2024-03-24
Accepted: 2024-03-29
Published Online: 2024-08-03
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2024-2038/html
Scroll to top button