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Surfaces with Vanishing Moebius Form in S n

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Abstract

An important Moebius invariant in the theory of Moebius surfaces in S n is the so-called Moebius form. In this paper, we give a complete classification of surfaces in S n with vanishing Moebius form under the Moebius transformation group.

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References

  1. Wang, C. P.: Moebius geometry of submanifolds in S n. Manuscripta Math., 96, 517–534 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Li, H., Wang, C. P., Wu, F.: A Moebius characterization of Veronese surfaces in S n. Math. Ann., 319, 707–714 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Willmore, T. J.: Surfaces in conformal geometry. Ann. Global Anal. Geom., 18, 255–264 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Weiner, J.: On a problem of Chen, Willmore, et al.. Indiana Univ. Math. J., 27, 19–35 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burstall, F., Pedit, F., Pinkall, U.: Schwarzian derivatives and flows of surfaces. arXiv. org/math. DG/0111169

  6. Bryant, R.: A duality theorem for Willmore surfaces. J. Differential Geom., 20, 23–53 (1984)

    MathSciNet  MATH  Google Scholar 

  7. Castro, I., Urbano F.: Willmore surfaces of R 4 and the Whitney sphere. Ann. Global Anal. Geom., 19, 153–175 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, H.: Willmore surfaces in S n. Ann. Global Anal. Geom., 21, 203–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Montiel, S.: Willmore two-spheres in the four-sphere. Trans. AMS, 352, 4469–4486 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pinkall, U.: Hopf tori in S 3. Invent. Math., 8, 379–386 (1985)

    Article  MathSciNet  Google Scholar 

  11. Hu, Z. J., Li, H.: Submanifolds with constant Moebius scalar curvature in S n. Manuscripta Math., 111, 287–302 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Li, H., Liu, H. L., Wang, C. P., Zhao, G. S.: Moebius isoparametric hypersurfaces in S n+1 with two distinct principal curvatures. Acta Math. Sinica, English Series, 18, 437–446 (2002)

    Article  MATH  Google Scholar 

  13. Liu, H. L., Wang, C. P., Zhao, G. S.: Moebius isotropic submanifolds in S n. Tohoku Math. J., 53, 553–569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan, 18, 380–385 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bryant, R.: Minimal surfaces of constant curvature in S n. Trans. AMS, 290, 259–271 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Wallach, N.: Extension of locally defined minimal immersions of spheres into spheres. Arch. Math., 21, 210–213 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kenmotsu, K.: On minimal immersions of R 2 into S. J. Math. Soc. Japan, 28, 182–191 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China

    Hai Zhong Li

  2. School of Mathematical Sciences (LMAM), Peking University, Beijing, 100871, P. R. China

    Chang Ping Wang

Authors
  1. Hai Zhong Li
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  2. Chang Ping Wang
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Correspondence to Hai Zhong Li.

Additional information

Hai Zhong Li: Partially supported the Alexander Humboldt Stiftung and Zhongdian Grant of NSFC

Chang Ping Wang: Partially supported by 973 Project, RFDP, Quishi Award and the Jiechu Grant of NSFC

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Li, H.Z., Wang, C.P. Surfaces with Vanishing Moebius Form in S n . Acta Math Sinica 19, 671–678 (2003). https://doi.org/10.1007/s10114-003-0309-8

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  • DOI: https://doi.org/10.1007/s10114-003-0309-8

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