Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ZERO MEAN CURVATURE SURFACES IN ISOTROPIC THREE-SPACE

  • Received : 2019.08.20
  • Accepted : 2020.05.07
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We examine the theory of surfaces in the isotropic three-space, with emphases on the surfaces related to the zero mean curvature.

Keywords

References

  1. R. Aiyama and K. Akutagawa, Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces in H3(-c2) and S31(c2), Ann. Global Anal. Geom. 17 (1999), no. 1, 49-75. https://doi.org/10.1023/A:1006504614150
  2. L. J. Alias 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
  3. R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Asterisque No. 154-155 (1987), 12, 321-347, 353 (1988).
  4. T. H. Colding and W. P. Minicozzi, II, Shapes of embedded minimal surfaces, Proc. Natl. Acad. Sci. USA 103 (2006), no. 30, 11106-11111. https://doi.org/10.1073/pnas.0510379103
  5. L. C. B. da Silva, Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces, preprint arXiv:1810.00080v3.
  6. M. Dede, C. Ekici, and A. C. Coken, On the parallel surfaces in Galilean space, Hacet. J. Math. Stat. 42 (2013), no. 6, 605-615.
  7. Y. W. Kim, S. Koh, H. Shin, and S. Yang, Generalized surfaces with constant H/K in Euclidean three-space, Manuscripta Math. 124 (2007), no. 3, 343-361. https://doi.org/10.1007/s00229-007-0125-z
  8. Y. W. Kim, S. Koh, H. Shin, and S. Yang, Spacelike maximal surfaces, timelike minimal surfaces, and Bjorling representation formulae, J. Korean Math. Soc. 48 (2011), no. 5, 1083-1100. https://doi.org/10.4134/JKMS.2011.48.5.1083
  9. Y. W. Kim and S.-D. Yang, Prescribing singularities of maximal surfaces via a singular Bjorling representation formula, J. Geom. Phys. 57 (2007), no. 11, 2167-2177. https://doi.org/10.1016/j.geomphys.2007.04.006
  10. O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space L3, 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
  12. H. Liu, Representation of surfaces in 3-dimensional lightlike cone, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 4, 737-748. http://projecteuclid.org/euclid.bbms/1320763134
  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
  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
  15. M. Pember, Weierstrass-type representations, Geom. Dedicata 204 (2020), 299-309. https://doi.org/10.1007/s10711-019-00456-y
  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.
  17. W. Rossman, M. Umehara, and K. Yamada, Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus, Tohoku Math. J. (2) 49 (1997), no. 4, 449-484. https://doi.org/10.2748/tmj/1178225055
  18. Y. Sato, d-minimal surfaces in three-dimensional singular semi-Euclidean space ℝ0,2,1, 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 3-space with applications, in Lorentzian geometry and related topics, 253-273, Springer Proc. Math. Stat., 211, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-66290-9_14
  22. M. Weber, On Karcher's twisted saddle towers, in Geometric analysis and nonlinear partial differential equations, 117-127, Springer, Berlin, 2003.
  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.