Generalizations of the Catenoid and the Helicoid

  • David E. Blair
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


In this lecture we will discuss various generalizations of the catenoid and the helicoid as well as related differential geometric notions including minimality, quasi-umbilicity and conformal flatness.


Minimal Surface Normal Bundle Lagrangian Submanifolds Minimal Hypersurface Ricci Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bang, K., Blair, D.E.: The Schouten tensor and conformally flat manifolds. In: Topics in Differential Geometry, pp. 1–28. Editura Academiei Române, Bucharest (2008)Google Scholar
  2. 2.
    Blair, D.E.: On a generalization of the catenoid. Canad. J. Math. 27, 231–236 (1975)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Blair, D.E.: Lagrangian helicoids. Mich. Math. J. 50, 187–200 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Blair, D.E.: On generalized catenoids. In: Contemporary Geometry and Related Topics, pp. 37–49. C̆igoja, Belgrade (2006)Google Scholar
  5. 5.
    Blair, D.E.: On Lagrangian catenoids. Canad. Math. Bull. 50, 321–333 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Blair, D.E., Vanstone, J.R.: A generalization of the helicoid. In: Proceedings of the U.S.-Japan Seminar on Minimal Submanifolds and Geodesics, pp. 13–16. Kaigai, Tokyo (1978)Google Scholar
  7. 7.
    Cartan, É.: La déformation des hypersurfaces dans l’espace conforme réel a n ≥ 5 dimensions. Bull. Soc. Math. France 45, 57–121 (1917)MATHMathSciNetGoogle Scholar
  8. 8.
    Castro, I.: The Lagrangian version of a theorem of J. B. Meusnier. In: Proceedings of the Summer School on Differential Geometry, pp. 83–89. Dep. de Matemática, Universidade de Coimbra (1999)Google Scholar
  9. 9.
    Castro, I., Urbano, F.: On a minimal Lagrangian submanifold of \(\mathbb{C}^{n}\) foliated by spheres. Mich. Math. J. 46, 301–321 (1999)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, B.-Y.: Geometry of Submanifolds. Dekker, New York (1973)MATHGoogle Scholar
  11. 11.
    Chen, B.-Y.: Representation of flat Lagrangian H-umbilical submanifolds in complex Euclidean spaces. Tô hok Math. J. 51, 13–20 (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Chen B.-Y.: Riemannian submanifolds. In: Handbook of Differential Geometry, vol. 1, pp. 187–418. Elsevier Science B. V., Amsterdam (2000)Google Scholar
  13. 13.
    Chen, B.-Y., Yano, K.: Sous-variété localemaent conformes à un espace euclidien. C. R. Acad. Sci. Paris 275, 123–126 (1972)MATHMathSciNetGoogle Scholar
  14. 14.
    Harvey, R., Lawson, H.B.: Calibrated geometries. Acta. Math. 148, 47–157 (1982)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Jagy, W.C.: Minimal hypersurfaces foliated by sphere. Mich. Math. J. 38, 255–270 (1991)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lumiste, Ü., Väljas, M.: On geometry of totally quasiumbillical submanifolds. Taru Riikl. Ü l. Toimetised No. 836, pp. 172–185 (1989)Google Scholar
  17. 17.
    Meusnier, J.B.: Mémoire sur la courbure des surfaces. Mémoires Math. Phys. 10, 477–510 (1785)Google Scholar
  18. 18.
    Moore, J.D., Morvan, J.M.: Sous-variété localemaent conformément plates de codimension quatre. C. R. Acad. Sci. Paris 287, 655–657 (1987)MathSciNetGoogle Scholar
  19. 19.
    Schouten, J.A.: Über die konforme Abbildung n-dimensionaler Mannigfaltigkeiten mit quadratischer Maßbestimmung; auf eine Mannigfaltigkeit mit euklidischer Maßbestimmung. Math. Z. 11, 58–88 (1921)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations