Mathematische Zeitschrift

, 259:827 | Cite as

Singularities of maximal surfaces

  • Shoichi Fujimori
  • Kentaro Saji
  • Masaaki Umehara
  • Kotaro Yamada


We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.


Maximal surfaces Minkowski space de Sitter space Cuspidal cross cap 

Mathematics Subject Classification (2000)

Primary 57R45 Secondary 53A10 53B20 


  1. 1.
    Arnol’d V.I., Gusein-Zade S.M. and Varchenko A.N. (1985). Singularities of differentiable maps, vol. 1. Monographs in Mathematics 82. Birkhäuser, Basel Google Scholar
  2. 2.
    Bryant R. (1987). Surfaces of mean curvature one in hyperbolic space. Astérisque 154–155: 321–347 Google Scholar
  3. 3.
    Cleave J.P. (1980). The form of the tangent developable at points of zero torsion on space curves. Math. Proc. Camb. Philos. Soc. 88: 403–407 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fernández I., López F.J. and Souam R. (2005). The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz–Minkowski space \({\mathbb{L}}^3\). Math. Ann. 332: 605–643 CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Fujimori S. (2006). Spacelike CMC 1 surfaces with elliptic ends in de Sitter 3-Space. Hokkaido Math. J. 35: 289–320 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fujimori, S., Rossman, W., Umehara, M., Yamada, K., Yang, S.-D.: Spacelike mean curvature one surfaces in de Sitter 3-space. Preprint, arXiv:0706.0973Google Scholar
  7. 7.
    Golubitsky M. and Guillemin V. (1973). Stable mappings and their singularities. Graduate Texts in Mathematics, vol. 14. Springer, Heidelberg Google Scholar
  8. 8.
    Ishikawa G. and Machida Y. (2006). Singularities of improper affine spheres and surfaces of constant Gaussian curvature. Int. J. Math. 17: 269–293 CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Izumiya, S., Saji, K., Takahashi, M.: Horospherical flat surfaces in hyperbolic 3-space. Preprint,
  10. 10.
    Izumiya S., Saji K. and Takeuchi N. (2007). Circular surfaces. Adv. Geometry 7: 295–313 CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Kobayashi O. (1984). Maximal surfaces with conelike singularities. J. Math. Soc. Jpn. 36: 609–617 zbMATHGoogle Scholar
  12. 12.
    Kokubu M., Rossman W., Saji K., Umehara M. and Yamada K. (2005). Singularities of flat fronts in hyperbolic 3-space. Pac. J. Math. 221: 303–351 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Saji, K., Umehara, M., Yamada, K.: The geometry of fronts, to appear in Ann. Math., math.DG/0503236Google Scholar
  14. 14.
    Umehara M. and Yamada K. (1993). Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space. Ann. Math. 137(2): 611–638 MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Umehara M. and Yamada K. (2006). Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35: 13–40 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Whitney H. (1944). The singularities of a smooth n-manifold in (2n − 1)-space. Ann. Math. 45: 247–293 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Shoichi Fujimori
    • 1
  • Kentaro Saji
    • 2
  • Masaaki Umehara
    • 3
  • Kotaro Yamada
    • 4
  1. 1.Department of MathematicsFukuoka University of EducationMunakata, FukuokaJapan
  2. 2.Department of MathematicsHokkaido UniversitySapporoJapan
  3. 3.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan
  4. 4.Faculty of MathematicsKyushu UniversityHigashi-ku, FukuokaJapan

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