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Zero mean curvature surfaces in isotropic three-space. (English) Zbl 1476.53036

The article discusses the space \(\mathbb I^3\) (isotropic three space) and zero mean curvature surfaces in \(\mathbb I^3\). In particular, in this article, authors discuss many aspects of ZMC surfaces in \(\mathbb I^3\), such as Weierstrass representation, Björling representation, interpretation of Weierstrass data, surfaces with umbilicial points, etc.
Extensive details and directions to work on many other aspects have been suggested.

MSC:

53A35 Non-Euclidean differential geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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[1] R. Aiyama and K. Akutagawa,Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces inH3(−c2)andS31(c2), Ann. Global Anal. Geom.17 (1999), no. 1, 49-75.https://doi.org/10.1023/A:1006504614150 · Zbl 0948.53032
[2] L. J. Al´ıas and B. Palmer,Curvature properties of zero mean curvature surfaces in fourdimensional Lorentzian space forms, Math. Proc. Cambridge Philos. Soc.124(1998), no. 2, 315-327.https://doi.org/10.1017/S0305004198002618 · Zbl 0947.53031
[3] R. L. Bryant,Surfaces of mean curvature one in hyperbolic space, Ast´erisque No. 154155 (1987), 12, 321-347, 353 (1988).
[4] T. H. Colding and W. P. Minicozzi, II,Shapes of embedded minimal surfaces, Proc. Natl. Acad. Sci. USA103(2006), no. 30, 11106-11111.https://doi.org/10.1073/pnas. 0510379103 · Zbl 1175.53008
[5] L. C. B. da Silva,Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces, preprint arXiv:1810.00080v3. · Zbl 1468.53005
[6] M. Dede, C. Ekici, and A. C. C¸ ¨oken,On the parallel surfaces in Galilean space, Hacet. J. Math. Stat.42(2013), no. 6, 605-615. · Zbl 1303.53020
[7] Y. W. Kim, S. Koh, H. Shin, and S. Yang,Generalized surfaces with constantH/K in Euclidean three-space, Manuscripta Math.124(2007), no. 3, 343-361.https://doi. org/10.1007/s00229-007-0125-z · Zbl 1167.53027
[8] ,Spacelike maximal surfaces, timelike minimal surfaces, and Bj¨orling representation formulae, J. Korean Math. Soc.48(2011), no. 5, 1083-1100.https://doi.org/ 10.4134/JKMS.2011.48.5.1083 · Zbl 1250.53016
[9] Y. W. Kim and S.-D. Yang,Prescribing singularities of maximal surfaces via a singular Bj¨orling representation formula, J. Geom. Phys.57(2007), no. 11, 2167-2177.https: //doi.org/10.1016/j.geomphys.2007.04.006 · Zbl 1141.53012
[10] O. Kobayashi,Maximal surfaces in the3-dimensional Minkowski spaceL3, Tokyo J. Math.6(1983), no. 2, 297-309.https://doi.org/10.3836/tjm/1270213872
[11] H. Liu,Surfaces in the lightlike cone, J. Math. Anal. Appl.325(2007), no. 2, 1171-1181. https://doi.org/10.1016/j.jmaa.2006.02.064 · Zbl 1108.53011
[12] ,Representation of surfaces in 3-dimensional lightlike cone, Bull. Belg. Math. Soc. Simon Stevin18(2011), no. 4, 737-748.http://projecteuclid.org/euclid.bbms/ 1320763134 · Zbl 1242.53024
[13] H. Liu and S. D. Jung,Hypersurfaces in lightlike cone, J. Geom. Phys.58(2008), no. 7, 913-922.https://doi.org/10.1016/j.geomphys.2008.02.011 · Zbl 1144.53028
[14] X. Ma, C. Wang, and P. Wang,Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space, Adv. Math.249(2013), 311-347.https: //doi.org/10.1016/j.aim.2013.09.013 · Zbl 1288.53051
[15] M. Pember,Weierstrass-type representations, Geom. Dedicata204(2020), 299-309. https://doi.org/10.1007/s10711-019-00456-y · Zbl 1433.53015
[16] H. Pottman and Y. Liu,Discrete Surfaces in isotropic geometry, R. Martin, M. Sabin, J. Winkler (Eds.): Mathematics of Surfaces 2007, LNCS 4647, pp. 341-363, 2007. · Zbl 1163.68359
[17] W. Rossman, M. Umehara, and K. Yamada,Irreducible constant mean curvature1 surfaces in hyperbolic space with positive genus, Tohoku Math. J. (2)49(1997), no. 4, 449-484.https://doi.org/10.2748/tmj/1178225055 · Zbl 0913.53025
[18] Y. Sato,d-minimal surfaces in three-dimensional singular semi-Euclidean spaceR0,2,1, arXiv:1809.07518v1.
[19] J. J. Seo,On the geometry of curves and surfaces in the isotropic three-space, Ph.D. thesis (2020), Korea University.
[20] J. J. Seo and S.-D. Yang,Constant ratio curves in the isotropic plane and their deflection properties, J. Korean Soc. Math. Edu. See. B: Pure Appl. Math., to appear.
[21] M. Umehara and K. Yamada,Surfaces with light-like points in Lorentz-Minkowski 3space with applications, in Lorentzian geometry and related topics, 253-273, Springer Proc. Math. Stat., 211, Springer, Cham, 2017.https://doi.org/10.1007/978-3-31966290-9_14 · Zbl 1402.53007
[22] M. Weber,On Karcher’s twisted saddle towers, in Geometric analysis and nonlinear partial differential equations, 117-127, Springer, Berlin, 2003. · Zbl 1043.53012
[23] I. M. Yaglom,A Simple Non-Euclidean Geometry and Its Physical Basis, translated from the Russian by Abe Shenitzer, Springer-Verlag, New York, 1979 · Zbl 0393.51013
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