Skip to main content
SpringerLink
Log in

Bernstein type theorems with flat normal bundle

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

Abstract

We prove Bernstein type theorems for minimal n-submanifolds in ℝn+p with flat normal bundle. Those are natural generalizations of the corresponding results of Ecker-Huisken and Schoen-Simon-Yau for minimal hypersurfaces.

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. Cheng, S.Y., Li, P., Yau, S.-T.: Heat equations on minimal submanifolds and their applications. Amer. J. Math. 106(5), 1033–1065 (1984)

    MathSciNet  Google Scholar 

  2. Ecker, K., Huisken, G.: A Bernstein result for minimal graphs of controlled growth. J. Differential Geom. 31(2), 397–400 (1990)

    MathSciNet  Google Scholar 

  3. Fischer-Colbrie, D.: Some rigidity theorems for minimal submanifolds of the sphere. Acta Math. 145(1–2), 29–46 (1980)

    Google Scholar 

  4. Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33(2), 199–211 (1980)

    MathSciNet  Google Scholar 

  5. Hildebrandt, S., Jost, J., Widman, K.-O.: Harmonic mappings and minimal submanifolds. Invent. Math. 62(2), 269–298 (1980/81)

    Article  MathSciNet  Google Scholar 

  6. Hoffman, D.A., Osserman, R., Schoen, R.: On the Gauss map of complete surfaces of constant mean curvature in R3 and R4. Comment. Math. Helv. 57(4), 519–531 (1982)

    MathSciNet  Google Scholar 

  7. Hsiang, W.-Y., Palais, R.S., Terng, C.-L.: The topology of isoparametric submanifolds. J. Differential Geom. 27(3), 423–460 (1988)

    MathSciNet  Google Scholar 

  8. Jost, J., Xin, Y.L.: Bernstein type theorems for higher codimension. Calc. Var. Partial Differential Equations 9(4), 277–296 (1999)

    Article  MathSciNet  Google Scholar 

  9. Lawson, Jr. H.B., Osserman, R.: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math. 139(1–2), 1–17 (1977)

    Google Scholar 

  10. Ni, L.: Gap theorems for minimal submanifolds in \({\bf R}\sp {n+1}\). Comm. Anal. Geom. 9(3), 641–656 (2001)

    MathSciNet  Google Scholar 

  11. Nitsche, J.C.C.: Lectures on minimal surfaces. Vol. 1. Cambridge University Press, Cambridge, 1989. Introduction, fundamentals, geometry and basic boundary value problems, Translated from the German by Jerry M. Feinberg, With a German foreword

  12. Simons, J.: Minimal varieties in riemannian manifolds. Ann. of Math. (2) 88, 62–105 (1968)

    MATH  MathSciNet  Google Scholar 

  13. Schoen, R., Simon, L., Yau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134(3–4), 275–288 (1975)

    MathSciNet  Google Scholar 

  14. Smoczyk, K., Wang, G., Xin, Y.L.: Mean curvature flow with flat normal bundles. arXiv:math.DG/0411010, v1 Oct. 31, 2004

  15. Terng, C.-L.: Isoparametric submanifolds and their Coxeter groups. J. Differential Geom. 21(1), 79–107 (1985)

    MATH  MathSciNet  Google Scholar 

  16. Terng, C.-L.: Convexity theorem for isoparametric submanifolds. Invent. Math. 85(3), 487–492 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  17. Terng, C.-L.: Submanifolds with flat normal bundle. Math. Ann. 277(1), 95–111 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang, M.-T.: On graphic Bernstein type results in higher codimension. Trans. Amer. Math. Soc. 355(1), 265–271 (electronic), (2003)

    Google Scholar 

  19. Wang, M.-T.: Stability and curvature estimates for minimal graphs with flat normal bundles, arXiv:math.DG/0411169, v1 Nov. 8 and v2 Nov. 11, 2004

  20. Xin, Y.L.: Bernstein type theorems without graphic condition. preprint 2003, to appear in Asian J. Math.

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences Inselstr. 22-26, 04103, Leipzig, Germany

    Knut Smoczyk

  2. Institute of Mathematics, Fudan University Shanghai, China and Max Planck Institute for Mathematics in the Sciences Inselstr. 22-26, 04103, Leipzig, Germany

    Guofang Wang

  3. Institute of Mathematics Fudan University, Shanghai, China

    Y. L. Xin

Authors
  1. Knut Smoczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guofang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. L. Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Knut Smoczyk.

Additional information

The research of the first author was supported by a Heisenberg fellowship of the DFG

The research of the third author was partially supported by project # 973 of MSTC and SFECC

Mathematics Subject Classification Primary (2000) 53C42

Rights and permissions

Reprints and permissions

About this article

Cite this article

Smoczyk, K., Wang, G. & Xin, Y.L. Bernstein type theorems with flat normal bundle. Calc. Var. 26, 57–67 (2006). https://doi.org/10.1007/s00526-005-0359-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-005-0359-0

Keywords

Search

Navigation