Skip to main content

Minimal Submanifolds

  • Chapter
  • First Online:
  • 1610 Accesses

Part of the book series: Universitext ((UTX))

Abstract

The theory of minimal submanifolds is one of the most beautiful and developed subjects of differential geometry. The aim of this chapter is to introduce a few of its general aspects.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Barbosa, L., Delgado, J.: Ruled submanifolds in spaces forms with mean curvature of nonzero constant length. Am. J. Math. 106, 763–780 (1984)

    Article  Google Scholar 

  2. Barbosa, L., do Carmo, M.: A necessary condition for a metric in M n to be minimally immersed in \(\mathbb {R}^{n+1}\). An. Acad. Brasil Ciênc. 50, 451–454 (1978)

    Google Scholar 

  3. Barbosa, L., Dajczer, M., Jorge, L.: Rigidity of minimal immersions in spaces forms. Math. Ann. 267, 433–437 (1984)

    Article  MathSciNet  Google Scholar 

  4. Barbosa, L., Dajczer, M., Jorge, L.: Minimal ruled submanifolds in spaces forms of constant curvature. Indiana Univ. Math. J. 33, 531–547 (1984)

    Article  MathSciNet  Google Scholar 

  5. Berger, M., Gauduchon, P., Mazet, E.: Le Spectre d’une Varieté Riemannienne. Lecture Notes in Mathematics, vol. 194. Springer, New York (1971)

    Chapter  Google Scholar 

  6. Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Differ. Geom. 1, 111–125 (1967)

    Article  MathSciNet  Google Scholar 

  7. Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60, 568–578 (1993)

    Article  MathSciNet  Google Scholar 

  8. Chen, B.-Y.: A Riemannian invariant and its applications to submanifold theory. Results Math. 27, 17–26 (1995)

    Article  MathSciNet  Google Scholar 

  9. Chen, B.-Y.: Some new obstruction to minimal and Lagrangian isometric immersions. Jpn. J. Math. 26, 105–127 (2000)

    Article  MathSciNet  Google Scholar 

  10. Chen, B.-Y.: On isometric minimal immersions from warped products into real space forms. Proc. Edinb. Math. Soc. 45, 579–587 (2002)

    Article  MathSciNet  Google Scholar 

  11. Chern, S.S., Osserman, R.: Remarks on the Riemannian metric of a minimal submanifold. In: Geometry Symposium, Utrecht 1980 (Utrecht, 1980). Lecture Notes in Mathematics, vol. 894, pp. 49–90. Springer, New York (1981)

    Google Scholar 

  12. Costa, E.: Codimension two submanifolds with 2-nonnegative curvature operator. Arch. Math. 90, 82–86 (2008)

    Article  MathSciNet  Google Scholar 

  13. Dajczer, M., Florit, L.: On Chen’s basic equality. Ill. J. Math. 42, 97–106 (1998)

    Article  MathSciNet  Google Scholar 

  14. Dajczer, M., Gromoll, D.: Gauss parametrization and rigidity aspects of submanifolds. J. Differ. Geom. 22, 1–12 (1985)

    Article  MathSciNet  Google Scholar 

  15. Dajczer, M., Rodríguez, L.: On asymptotic directions of minimal immersions. Math. Z. 176, 187–194 (1981)

    Article  MathSciNet  Google Scholar 

  16. Dajczer, M., Vlachos, Th.: The associated family of an elliptic surface and an application to minimal submanifolds. Geom. Dedicata 178, 259–275 (2015)

    Article  MathSciNet  Google Scholar 

  17. Dajczer, M., Onti, C.-R., Vlachos, Th.: Einstein submanifolds with flat normal bundle in space forms are holonomic. Proc. Am. Math. Soc. 146, 4035–4038 (2018)

    Article  MathSciNet  Google Scholar 

  18. De Turck, D., Ziller, W.: Minimal isometric immersions of spherical space forms into spheres. Comment. Math. Helv. 67, 428–458 (1992)

    Article  MathSciNet  Google Scholar 

  19. do Carmo, M., Dajczer, M.: Necessary and sufficient conditions for existence of minimal hypersurfaces in spaces of constant curvature. Bol. Soc. Brasil. Mat. 12, 113–121 (1981)

    Article  MathSciNet  Google Scholar 

  20. do Carmo, M., Wallach, N.: Minimal immersions of spheres into spheres. Ann. Math. 93, 43–62 (1971)

    Article  MathSciNet  Google Scholar 

  21. Fialkow, A.: Hypersurfaces of a space of constant curvature. Ann. Math. 39, 762–785 (1938)

    Article  MathSciNet  Google Scholar 

  22. Freitas, G.: Submanifolds with homothetic Gauss map in codimension two. Geom. Dedicata, 151–170 (2016)

    Google Scholar 

  23. Hasanis, Th., Vlachos, Th.: Ricci curvature and minimal submanifolds. Pac. J. Math. 197, 13–24 (2001)

    Article  MathSciNet  Google Scholar 

  24. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Interscience Publishers, New York (1969)

    MATH  Google Scholar 

  25. Lawson, B.: Complete minimal surfaces in S 3. Ann. Math. 92, 335–374 (1970)

    Article  MathSciNet  Google Scholar 

  26. Lawson, B.: Lectures on Minimal Submanifolds. Mathematics Lecture Series, vol. 9. Publish or Perish, Inc., Wilmington (1980)

    Google Scholar 

  27. Matsuyama, Y.: Minimal Einstein submanifolds with codimension two. Tensor (N.S.) 52, 61–68 (1993)

    Google Scholar 

  28. Pinl, M., Ziller, W.: Minimal hypersurfaces in spaces of constant curvature. J. Differ. Geom. 11, 335–343 (1976)

    Article  MathSciNet  Google Scholar 

  29. Ricci, G.: Opere. Edizione Cremonese, Rome (1956)

    Google Scholar 

  30. Ryan, P.: Homogeneity and some curvature conditions for hypersurfaces. Tôhoku Math. J. 21, 363–388 (1969)

    Article  MathSciNet  Google Scholar 

  31. Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 18, 380–385 (1966)

    Article  MathSciNet  Google Scholar 

  32. Toth, G.: Eigenmaps and the space of minimal immersions between spheres. Indiana Univ. Math. J. 46, 637–658 (1997)

    Article  MathSciNet  Google Scholar 

  33. Vlachos, T.: Intrinsic obstructions to the existence of isometric minimal immersions. Pac. J. Math. 205, 491–510 (2002)

    Article  MathSciNet  Google Scholar 

  34. Vlachos, T.: Almost-Einstein hypersurfaces in the Euclidean space. Ill. J. Math. 53, 1121–1235 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Xin, Y.: Minimal Submanifolds and Related Topics. Nankai Tracts in Mathematics, vol. 8. World Scientific Publishing Co., Inc., River Edge (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Science+Business Media, LLC, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dajczer, M., Tojeiro, R. (2019). Minimal Submanifolds. In: Submanifold Theory . Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9644-5_3

Download citation

Publish with us

Policies and ethics