Singular Minimal Surfaces which are Minimal

Year 2021, Volume: 4 Issue: 4, 136 - 146, 30.12.2021

Muhittin Evren Aydın , Ayla Erdur Kara , Mahmut Ergüt

https://doi.org/10.32323/ujma.984462

Cited By: 1

Abstract

In the present paper, we discuss the singular minimal surfaces in Euclidean $3-$space $\mathbb{R}^{3}$ which are minimal. Such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the usual condition of singular minimality by using a special semi-symmetric metric connection instead of the Levi-Civita connection on $\mathbb{R}^{3}$. With this new connection, we prove that, besides planes, the singular minimal surfaces which are minimal are the generalized cylinders, providing their explicit equations. A trivial outcome is observed when we use a special semi-symmetric non-metric connection. Furthermore, our discussion is adapted to the Lorentz-Minkowski 3-space.

Keywords

Minimal surface , Semi-symmetric connection , Singular minimal surface , Translation surface

References

• [1] N. S. Agashe and M.R. Chafle, A semi-symmetric non-metric connection on a Riemannian manifold, Indian J. Pure Appl. Math. 23(6) (1992), 399-409.
• [2] N. S. Agashe and M. R. Chafle, On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection, Tensor 55(2) (1994), 120-130.
• [3] M. A. Akyol and S. Beyendi, Riemannian submersion endowed with a semi-symmetric non-metric connection, Konuralp J. Math. 6(1) (2018), 188-193.
• [4] G. Ayar and D. Demirhan, ”Ricci solitons on Nearly Kenmotsu manifolds with semi-symmetric metric connection, J. Eng. Tech. Appl. Sci., 4(3) (2019), 131-–140.
• [5] M. E. Aydin, A. Erdur and M. Ergut, Singular minimal translation graphs in Euclidean spaces, J. Korean Math. Soc. 58(1) (2021), 109–122.
• [6] R. B¨ohme, S. Hildebrant and E. Taush, The two-dimensional analogue of the catenary, Pac. J. Math. 88(2) (1980), 247-278.
• [7] S. K. Chaubey and A. Yildiz, Riemannian manifolds admitting a new type of semisymmetric nonmetric connection, Turk. J. Math. 43(4) (2019), 1887-1904.
• [8] B.-Y. Chen, Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific, Hackensack, NJ, 2017.
• [9] J. G. Darboux, Th´eorie G´enerale des Surfaces, Livre I, Gauthier-Villars, Paris, 1914.
• [10] U. C. De and A. Barman, On a type of semisymmetric metric connection on a Riemannian manifold, Publ. Inst. Math., Nouv. S´er. 98 (112) (2015), 211-218.
• [11] U. Dierkes, A Bernstein result for enery minimizing hypersurfaces, Cal. Var. Part. Differ. Equ. 1(1) (1993), 37-54.
• [12] U. Dierkes, Singular minimal surfaces, Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, Heidelberg (2003), 176-193.
• [13] U. Dierkes, G. Huisken, The n-dimensional analogue of the catenary: existence and nonexistence, Pac. J. Math. 141 (1990), 47–54.
• [14] Y. Dogru, On some properties of submanifolds of a Riemannian manifold endowed with a semi-symmetric non-metric connection, An. S¸ t. Univ. Ovidius Constanta 19(3) (2011), 85-100.
• [15] K. Duggal, R. Sharma, Semi-symmetric metric connections in a semi-Riemannian manifold, Indian J. Pure Appl. Math. 17(11) (1986), 1276–1282.
• [16] A. Erdur, M. E. Aydin and M. Ergut, Singular minimal translation surfaces in Euclidean spaces endowed with semi-symmetric metric connections, arXiv:2010.16139 [math.DG].
• [17] Y. Fu and D. Yang, On constant slope spacelike surfaces in 3-dimensional Minkowski space, J. Math. Anal. Appl., 385(1) (2012), 208–220.
• [18] J. B. Gil, The catenary (almost) everywhere, Bolettin de la AMV XII(2) (2005), 251-258.
• [19] A. Gozutok and E. Esin, Tangent bundle of hypersurface with semi symmetric metric connection, Int. J. Contemp. Math. Sci. 7(6) (2012), 279-289.
• [20] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, 2nd Ed., 1998.
• [21] F. G¨uler, G. Saffak and E. Kasap, Timelike constant angle surfaces in Minkowski Space R31, Int. J. Contemp. Math. Sciences, 6(44) (2011), 2189-2200.
• [22] H. A. Hayden, Subspaces of a space with torsion, Proc. London Math. Soc. S2-34(1) (1932), 27-50.
• [23] D.T. Hieu, N. M. Hoang, Ruled minimal surfaces in R3 with density ez, Pacific J. Math. 243(2) (2009),, 277-285.
• [24] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8(1) (1999), 1-14.
• [25] T. Imai, Notes on semi-symmetric metric connections, Tensor 24 (1972), 293-296.
• [26] H. Liu and Y. Yu, Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad. 89(A) (2013), 111-113.
• [27] R. L´opez, Constant Mean Curvature Surfaces with Boundary, Springer-Verlag, Berlin (2013).
• [28] R. L´opez, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom. 7(1) (2014), 44-107.
• [29] R. L´opez, Separation of variables in equations of mean-curvature type, Proc. R. Soc. Edinb. Sect. A Math. 146(5) (2016), 1017-1035.
• [30] R. L´opez, Some geometric properties of translating solitons in Euclidean space, J. Geom. 109(40) (2018), 1-15.
• [31] R. L´opez, Invariant surfaces in Euclidean space with a log-linear density, Adv. Math. 339 (2018), 285-309.
• [32] R. L´opez, Invariant singular minimal surfaces, Ann. Glob. Anal. Geom. 53(4) (2018), 521-541.
• [33] R. L´opez, The Dirichlet problem for the α-singular minimal surface equation, Arch. Math. 112(2) (2019), 213-222.
• [34] R. L´opez, The two- dimensional analogue of the Lorentzian catenary and the Dirichlet problem, Pacific J. Math. 305(2) (2020), 693-719.
• [35] R. L´opez, Compact singular minimal surfaces with boundary, Amer. J. Math. 142(6) (2020), 1771–1795.
• [36] R. L´opez, The one dimensional case of the singular minimal surfaces with density, Geom. Dedicata 200 (2019), 303-320.
• [37] M.I. Munteanu, and A. I. Nistor, A new approach on constant angle surfaces in E3, Turk. J. Math. 33 (2009), 169-178.
• [38] C. Murathan and C. Ozg¨ur, Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions, Proc. Est. Acad. Sci. 57(4) (2008), 210-216.
• [39] Z. Nakao, Submanifolds of a Riemannian manifold with semisymmetric metric connections, Proc. Amer. Math. Soc. 54(1) (1976), 261-266.
• [40] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983.
• [41] F. Ozen Zengin, S. Altay Demirbag and S. A. Uysal, Some vector fields on a Riemannian manifold with semi-symmetric metric connection, Bull. Iranian Math. Soc. 38(2) (2012), 479-490.
• [42] C. Ozgur, On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection, Kuwait J. Sci. Eng. 37(2) (2010), 17-30.
• [43] H. F. Scherk, Bemerkungen ¨uber die kleinste Fl¨ache innerhalb gegebener Grenzen, J. Reine Angew. Math. 13 (1835), 185-208.
• [44] Y. Wang, Minimal translation surfaces with respect to semi–symmetric connections in R3 and R31 , arXiv:2003.12682v1 [math.DG].
• [45] D. Yang, J. Zhang and Y. Fu, A note on minimal translation graphs in Euclidean space, Mathematics 7(10) (2019).
• [46] K. Yano, On semi-symmetric metric connection, Rev. Roumaine Math. Pures Appl. 15 (1970), 1579-1586.
• [47] K. Yano and M. Kon, Structures on Manifolds, Series in Pure Math., World Scientific, 1984.
• [48] M. Yıldırım, On non-existence of weakly symmetric nearly Kenmotsu manifold with semi-symmetric metric connection, Konuralp J. Math. 9(2) (2021), 332–336.
• [49] A. Yucesan and N. Ayyildiz, Non-degenerate hypersurfaces of a semi-Riemannian manifold with a semi-symmetric metric connection, Arch. Math. (Brno) 44(1) (2008), 77–88.
• [50] A. Yucesan and E. Yasar, Non–degenerate hypersurfaces of a semi–Riemannian manifold with a semi–symmetric non–metric connection, Math. Rep. (Bucur) 14(64)(2) (2012), 209-219.