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Bernstein type theorems with flat normal bundle

  • Knut SmoczykEmail author
  • Guofang Wang
  • Y. L. Xin
Article

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.

Keywords

System Theory Natural Generalization Normal Bundle Type Theorem Minimal Hypersurface 
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References

  1. 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)MathSciNetGoogle Scholar
  2. 2.
    Ecker, K., Huisken, G.: A Bernstein result for minimal graphs of controlled growth. J. Differential Geom. 31(2), 397–400 (1990)MathSciNetGoogle Scholar
  3. 3.
    Fischer-Colbrie, D.: Some rigidity theorems for minimal submanifolds of the sphere. Acta Math. 145(1–2), 29–46 (1980)Google Scholar
  4. 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)MathSciNetGoogle Scholar
  5. 5.
    Hildebrandt, S., Jost, J., Widman, K.-O.: Harmonic mappings and minimal submanifolds. Invent. Math. 62(2), 269–298 (1980/81)CrossRefMathSciNetGoogle Scholar
  6. 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)MathSciNetGoogle Scholar
  7. 7.
    Hsiang, W.-Y., Palais, R.S., Terng, C.-L.: The topology of isoparametric submanifolds. J. Differential Geom. 27(3), 423–460 (1988)MathSciNetGoogle Scholar
  8. 8.
    Jost, J., Xin, Y.L.: Bernstein type theorems for higher codimension. Calc. Var. Partial Differential Equations 9(4), 277–296 (1999)CrossRefMathSciNetGoogle Scholar
  9. 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. 10.
    Ni, L.: Gap theorems for minimal submanifolds in \({\bf R}\sp {n+1}\). Comm. Anal. Geom. 9(3), 641–656 (2001)MathSciNetGoogle Scholar
  11. 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 forewordGoogle Scholar
  12. 12.
    Simons, J.: Minimal varieties in riemannian manifolds. Ann. of Math. (2) 88, 62–105 (1968)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Schoen, R., Simon, L., Yau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134(3–4), 275–288 (1975)MathSciNetGoogle Scholar
  14. 14.
    Smoczyk, K., Wang, G., Xin, Y.L.: Mean curvature flow with flat normal bundles. arXiv:math.DG/0411010, v1 Oct. 31, 2004Google Scholar
  15. 15.
    Terng, C.-L.: Isoparametric submanifolds and their Coxeter groups. J. Differential Geom. 21(1), 79–107 (1985)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Terng, C.-L.: Convexity theorem for isoparametric submanifolds. Invent. Math. 85(3), 487–492 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Terng, C.-L.: Submanifolds with flat normal bundle. Math. Ann. 277(1), 95–111 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 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. 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, 2004Google Scholar
  20. 20.
    Xin, Y.L.: Bernstein type theorems without graphic condition. preprint 2003, to appear in Asian J. Math.Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the Sciences Inselstr. 22-26LeipzigGermany
  2. 2.Institute of MathematicsFudan University Shanghai, China and Max Planck Institute for Mathematics in the Sciences Inselstr. 22-26LeipzigGermany
  3. 3.Institute of Mathematics Fudan UniversityShanghaiChina

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