Generalizations of the Catenoid and the Helicoid

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 
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  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

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