Minimals translation surfaces in a strict Walker 3-manifold

Yıl 2024, Cilt: 73 Sayı: 2, 554 - 568, 21.06.2024

Athoumane Nıang , Ameth Ndiaye , Abdoul Salam Diallo

https://doi.org/10.31801/cfsuasmas.1296991

Öz

In this paper, we study minimal translations surfaces in a strict Walker 3-manifold. Based on the existence of two isometries, we classify minimal translation surfaces on this class of manifold. Some drawings have been added to illustrate the shape of certain surfaces.

Anahtar Kelimeler

Minimal surfaces, translation surfaces, Walker manifolds

Kaynakça

• Brozos-Vazquez, M., Garcia-Rio, E., Gilkey, P., Nikevic, S., Vazquez-Lorenzo, R., The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics, 5. Morgan and Claypool Publishers, Williston, VT, 2009.
• Calvaruso, G., Van der Veken, J. Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field, J. Phys. A: Math. Theor., 43 (2010) 325207, 9 pp. https://doi.org/10.1088/1751-8113/43/32/325207
• Darboux, G. Lessons on the General Theory of Surfaces. I, II, Editions Jacques Gabay, Sceaux, 1993. 604 pp.
• Diallo, A. S., Ndiaye, A., Niang, A., Minimal Graphs on Three-Dimensional Walker Manifolds, Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, 425-438, Trends Math., Birkhauser/Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-57336-2-17
• Dillen, F., Verstraele, L., Zafindratafa, G., A Generalization of the Translation Surfaces of Scherk, In: Differential Geometry in Honor of Radu Rosca. K. U. L., (1991), 107-109.
• Inoguchi, I., Lopez, R., Munteanu, M. I., Minimal translation surfaces in the Heisenberg group Nil3, Geom. Dedicata, 16 (2012), 221-231. https://doi.org/10.1007/s10711-012-9702-8
• Liu, H., Yu, Y., Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad. Ser. A, 89 (2013), (9), 111-113. https://doi.org/10.3792/pjaa.89.111
• Lopez, R., Minimal translation surfaces in hyperbolic space, Beitrage Alge. Geom., 52 (2011), 105-112. https://doi.org/10.1007/s13366-011-0008-z
• Moruz, M., Munteanu, M. I., Minimal translation hypersurfaces in E4, J. Math. Analysis Appl., 439 (2016), 798-812. https://doi.org/10.1016/j.jmaa.2016.02.077
• Niang, A., Ndiaye, A., Diallo, A. S., A Classification of Strict Walker 3-Manifold, Konuralp J. Math., 9(1) (2021), 148-153.
• Scherk, H. F., Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen, J. Reine Angew. Math., 13 (1835), 185-208.
• Yang, D., Fu, Y., On affine translation surfaces in affine space, J. Math. Analysis Appl., 440 (2016) 437-450. https://doi.org/10.1016/j.jmaa.2016.03.066
• Yoon, D. W., Lee, C. W., Karacan, M. K., Some translation surfaces in the 3- dimensional Heisenberg group, Bull. Korean Math. Soc., 50(4) (2013), 1329-1343. https://doi.org/10.4134/BKMS.2013.50.4.1329
• Yoon, D. W., Weighted minimal translation surfaces in the Galilean space with density, Open Math., 15 (1) (2017), 459-466. https://doi.org/10.1515/math-2017-0043
• Walker, A. G., Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math., Oxford, 1(1) (1950), 69-79. https://doi.org/10.1093/qmath/1.1.69