Skip to main content
SpringerLink
Log in

A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with |A|2 ≤ 1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

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. Abresch U., Langer J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23(2), 175–196 (1986)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Cao L.F., Li H.: r-minimal submanifolds in space forms. Ann. Glob. Anal. Geom. 32, 311–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao H.-D., Zhou D.: On complete gradient shrinking Ricci solitons. J. Differ. Geom. 85, 175–186 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Colding T.H., Minicozzi W.P. II: Generic mean curvature flow I: generic singularities. Ann. Math. 175(2), 755–833 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding, Q., Wang, Z.: On the self-shrinking systems in arbitrary codimension spaces. arXiv:1012.0429v1

  7. Ding, Q., Xin, Y.L.: Volume growth, eigenvalue and compactness for self-shrinkers. arXiv: 1101.1411v1

  8. Ecker K., Huisken G.: Mean curvature evolution of entire graphs. Ann. Math. 130, 453–471 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huisken G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285–299 (1990)

    MathSciNet  MATH  Google Scholar 

  10. Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. In: Differential geometry: partial differential equations on manifolds. Proceedings of Symposia in Pure Mathematics, vol. 54, pp. 175–191. American Mathematical Society, Providence (1993)

  11. Le N.Q., Sesum N.: Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Commun. Anal. Geom. 19(4), 1–27 (2011)

    MathSciNet  Google Scholar 

  12. Li H.: Willmore submanifolds in a sphere. Math. Res. Lett. 9, 771–790 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Schoen R., Simon L., Yau S.-T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Simons J.: Minimal varieties in riemannian manifolds. Ann. Math. 88, 62–105 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  15. Smoczyk K.: Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 48, 2983–3004 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang L.: A Benstein type theorem for self-similar shrinkers. Geom. Dedicata 15, 297–303 (2011)

    Article  Google Scholar 

  17. Wang M.-T.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57, 301–338 (2001)

    MATH  Google Scholar 

  18. Yau S.-T.: Submanifolds with constant mean curvature. Am. J. Math. 96, 346–366 (1974)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    Huai-Dong Cao

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Haizhong Li

Authors
  1. Huai-Dong Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haizhong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haizhong Li.

Additional information

Communicated by J. Jost.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cao, HD., Li, H. A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879–889 (2013). https://doi.org/10.1007/s00526-012-0508-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-012-0508-1

Mathematics Subject Classification (2000)

Search

Navigation