Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

TRANSLATION AND HOMOTHETICAL SURFACES IN EUCLIDEAN SPACE WITH CONSTANT CURVATURE

  • Lopez, Rafael (Departamento de Geometria y Topologia Universidad de Granada) ;
  • Moruz, Marilena (Department of Mathematics Al. I. Cuza University of Iasi)
  • Received : 2014.08.11
  • Published : 2015.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

We study surfaces in Euclidean space which are obtained as the sum of two curves or that are graphs of the product of two functions. We consider the problem of finding all these surfaces with constant Gauss curvature. We extend the results to non-degenerate surfaces in Lorentz-Minkowski space.

Keywords

References

  1. J. L. M. Barbosa and A. Gervasio Colares, Minimal Surfaces in $R^3$, Lecture Notes in Mathematics, 1195, Springer-Verlag, Berlin, 1986.
  2. J. G. Darboux, Theorie Generale des Surfaces, Livre I, Gauthier-Villars, Paris, 1914.
  3. F. Dillen, I. Van de Woestyne, L. Verstraelen, and J. T. Walrave, The surface of Scherk in $E^3$ : a special case in the class of minimal surfaces defined as the sum of two curves, Bull. Inst. Math. Acad. Sin. 26 (1998), no. 4, 257-267.
  4. L. Jiu and H. Sun, On minimal homothetical hypersurfaces, Colloq. Math. 109 (2007), no. 2, 239-249. https://doi.org/10.4064/cm109-2-6
  5. H. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), no. 1-2, 141-149. https://doi.org/10.1007/BF01229219
  6. R. Lopez, Differential Geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom. 7 (2014), no. 1, 44-107.
  7. J. C. C. Nitsche, Lectures on Minimal Surfaces. Vol. 1, Cambridge University Press, Cambridge, 1989.
  8. H. F. Scherk, Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen, J. Reine Angew. Math. 13 (1835), 185-208.
  9. I. Van de Woestyne, A new characterization of the helicoids, Geometry and topology of submanifolds, V (Leuven/Brussels, 1992), 267-273, World Sci. Publ., River Edge, NJ, 1993.
  10. I. Van de Woestyne, Minimal homothetical hypersurfaces of a semi-Euclidean space, Results Math. 27 (1995), no. 3-4, 333-342. https://doi.org/10.1007/BF03322837

Cited by

  1. Translation Hypersurfaces with Constant Sr Curvature in the Euclidean Space vol.88, pp.4, 2016, https://doi.org/10.1590/0001-3765201620150326