Semiumbilic points for minimal surfaces in Euclidean \(4\)-space

Geometriae Dedicata volume 170, pages1–7(2014)Cite this article

Abstract

For a minimal surface in the Euclidean 4-space, a semiumbilic point is at which its normal curvature vanishes. We prove that the semiumbilic points are isolated, if the normal curvature neither changes its sign nor vanishes identically. We also show that any entire minimal surface with flat normal bundle is an affine plane.

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References

  1. 1.

    Aiyama, R., Akutagawa, K.: Surfaces with inflection points in Euclidean 4-space, submitted (2013)

  2. 2.

    Almgren Jr., F.J.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. 84, 277–292 (1966)

    Article  Google Scholar 

  3. 3.

    Bombieri, E., de Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    de Giorgi, E.: Una extensione del teorema di Bernstein. Ann. Scuola Norm. Sup. Pisa 19, 79–85 (1965)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Fleming, W.H.: On the oriented Plateau problem. Rend. Circ. Mat. Palermo 11, 69–90 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. 2nd edn, Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1983)

    Google Scholar 

  7. 7.

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

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Hoffman, D.A., Osserman, R.: The Gauss map of surfaces in \(\mathbb{R}^3\) and \(\mathbb{R}^4\). Proc. Lond. Math. Soc. 50, 27–56 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

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

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Jost, J., Xin, Y.L.: A Bernstein type theorem for special Lagrangian graph. Calc. Var. Partial Differ. Equ. 15, 299–312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Little, J.A.: On singularities of submanifolds of higher dimensional Euclidean spaces. Ann. Math. Pura Appl. (Ser. 4A) 83, 261–335 (1969)

    Article  MathSciNet  Google Scholar 

  12. 12.

    Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: The geometry of surfaces in \(4\)-space from a contact viewpoint. Geometriae Dedicata 54, 323–332 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: Some global properties of surfaces in \(4\)-space, In: Proceedings of 1st International Meeting on Geometry and Topology, Braga (Portugal), Public. Centro de Mathemática de Universidade do Minho, pp. 175–183 (1998)

  14. 14.

    Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

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

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Smoczyk, K., Wang, G., Xin, Y.L.: Bernstein type theorems with flat normal bundle. Calc. Var. Partial Differ. Equ. 26, 57–67 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

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

    Article  MATH  Google Scholar 

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Author information

Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan

    Reiko Aiyama

  2. Department of Mathematics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan

    Kazuo Akutagawa

Authors
  1. Reiko Aiyama
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  2. Kazuo Akutagawa
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Corresponding author

Correspondence to Reiko Aiyama.

Additional information

Kazuo Akutagawa is supported in part by the Grant-in-Aid for Challenging Exploratory Research, Japan Society for the Promotion of Science, No. 24654009.

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Aiyama, R., Akutagawa, K. Semiumbilic points for minimal surfaces in Euclidean \(4\)-space. Geom Dedicata 170, 1–7 (2014). https://doi.org/10.1007/s10711-013-9865-y

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Keywords

  • Semiumbilic points
  • Minimal surfaces in Euclidean 4-space
  • Normal curvature
  • Bernstein type theorem

Mathematics Subject Classification

  • 53A05
  • 53A10