Skip to main content

Advertisement

SpringerLink
Log in

On the DDVV conjecture and the comass in calibrated geometry (I)

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this paper, we proved the first non-trivial case of the DDVV conjecture. Namely, for all 3 × 3 matrices, the DDVV inequality is valid. We also classified all the minimal submanifolds for which the equality holds.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bryant R.L. (1991). Some remarks on the geometry of austere manifolds. Bol. Soc. Brasil. Mat. (N.S.) 21(2): 133–157

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen B.-Y. (1996). Mean curvature and shape operator of isometric immersions in real-space-forms. Glasgow Math. J. 38(1): 87–97

    Article  MATH  MathSciNet  Google Scholar 

  3. Chern, S.S., do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length. In Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), pp. 59–75. Springer, New York (1970)

  4. Dadok J. and Harvey R. (1999). The Pontryagin 4-form. Proc. Am. Math. Soc. 127(11): 3175–3180

    Article  MATH  MathSciNet  Google Scholar 

  5. Dajczer M. and Florit L.A. (2001). A class of austere submanifolds. Illinois J. Math. 45(3): 735–755

    MATH  MathSciNet  Google Scholar 

  6. De Smet P.J., Dillen F., Verstraelen L. and Vrancken L. (1999). A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35(2): 115–128

    MATH  MathSciNet  Google Scholar 

  7. Dillen, F., Fastenakels, J., Veken, J.: Remarks on an inequality involving the normal scalar curvature. DG/0610721 (2006)

  8. Gluck H., Mackenzie D. and Morgan F. (1995). Volume-minimizing cycles in Grassmann manifolds. Duke Math. J. 79(2): 335–404

    Article  MATH  MathSciNet  Google Scholar 

  9. Gu W. (1998). The stable 4-dimensional geometry of the real Grassmann manifolds. Duke Math. J. 93(1): 155–178

    Article  MATH  MathSciNet  Google Scholar 

  10. Harvey R. and Lawson H.B. (1982). Calibrated geometries. Acta Math. 148: 47–157

    Article  MATH  MathSciNet  Google Scholar 

  11. Lu, Z.: Proof of the normal scalar curvature conjecture, preprint

  12. Suceavă, B.: Some remarks on B. Y. Chen’s inequality involving classical invariants. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.). 45(2), 405–412 (2000), 1999

  13. Suceavă, B.D.: DDVV conjecture. preprint

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, UC Irvine, Irvine, CA, 92697, USA

    Timothy Choi & Zhiqin Lu

Authors
  1. Timothy Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiqin Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiqin Lu.

Additional information

The second author is partially supported by NSF Career award DMS-0347033 and the Alfred P. Sloan Research Fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Choi, T., Lu, Z. On the DDVV conjecture and the comass in calibrated geometry (I). Math. Z. 260, 409–429 (2008). https://doi.org/10.1007/s00209-007-0281-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-007-0281-6

Keywords

Search

Navigation