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Wintgen ideal submanifolds: reduction theorems and a coarse classification

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Abstract

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms.

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References

  1. Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal geometry of surfaces in the 4-sphere and quaternions Lecture Notes in Mathematics, vol. 1772. Springer (2002)

  2. Chen, B.Y.: Classification of Wintgen ideal surfaces in Euclidean \(4\)-space with equal Gauss and normal curvatures. Ann. Glob. Anal. Geom. 38, 145–160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Choi, T., Lu, Z.: On the DDVV conjecture and the comass in calibrated geometry (I). Math. Z. 260, 409–429 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dajczer, M., Tojeiro, R.: A class of austere submanifolds. Ill. J. Math. 45(3), 735–755 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Dajczer, M., Tojeiro, R.: All superconformal surfaces in \(R^4\) in terms of minimal surfaces. Math. Z. 261(4), 869–890 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dajczer, M., Tojeiro, R.: Submanifolds of codimension two attaining equality in an extrinsic inequality. Math. Proc. Camb. Philos. Soc. 146(2), 461–474 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Smet, P.J., Dillen, F., Verstraelen, L., Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35, 115–128 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Dillen, F., Fastenakels, J., Van Der Veken, J.: Remarks on an inequality involving the normal scalar curvature. In: Proceedings of the International Congress on Pure and Applied Differential Geometry-PADGE Brussels, pp. 83–92. Shaker Verlag, Aachen (2007)

  9. Ferus, D.: On the completeness of nullity foliations. Mich. Math. J. 18, 61–64 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ge, J., Tang, Z.: A proof of the DDVV conjecture and its equality case. Pac. J. Math. 237, 87–95 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guadalupe, I., Rodríguez, L.: Normal curvature of surfaces in space forms. Pac. J. Math. 106, 95–103 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, T., Ma, X., Wang, C., Xie, Z.: New examples and classification of Möbius homogeneous Wintgen ideal submanifolds. in preparation

  13. Li, T., Ma, X., Wang, C., Xie, Z.: Wintgen ideal submanifolds of codimension two, complex curves, and Möbius geometry. Tohoku Math. J. 68(4), 621–638 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, T., Ma, X., Wang, C.: Deformation of hypersurfaces preserving the möbius metric and a reduction theorem. Adv. Math. 256, 156–205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, T., Ma, X., Wang, C.: Wintgen ideal submanifolds with a low-dimensional integrable distribution. Front. Math. China 10(1), 111–136 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, H., Wang, C., Zhao, G.: Möbius isotropic submanifolds in \(S^n\). Tohoku Math. J. 53, 553–569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lu, Z.: Normal scalar curvature conjecture and its applications. J. Funct. Anal. 261, 1284–1308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ma, X., Xie, Z.X.: The Möbius geometry of Wintgen ideal submanifolds. In: ICM 2014 Satellite Conference on Real and Complex Submanifolds, Springer Proceedings in Mathematics & Statistics, vol. 106, pp. 411–425 (2014)

  19. Maltz, R.: Isometric immersions into spaces of constant curvature. Ill. J. Math. 15, 490–502 (1971)

    MathSciNet  MATH  Google Scholar 

  20. Petrovié-torgas̆ev, M., Verstraelen, L.: On Deszcz symmetries of Wintgen ideal submanifolds. Arch. Math. 44, 57–67 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Van der Veken, J.: Ideal Lagrangian submanifolds. to appear in Recent Advances in the Geometry of Submanifolds, Contemporary Mathematics, American Mathematical Society

  22. Wang, C.: Möbius geometry of submanifolds in \(S^n\). Manuscr. Math. 96, 517–534 (1998)

    Article  Google Scholar 

  23. Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993–995 (1979)

    MathSciNet  MATH  Google Scholar 

  24. Xie, Z.X.: Wintgen ideal submanifolds with vanishing Möbius form. Ann. Glob. Anal. Geom. 48, 331–343 (2015)

    Article  MATH  Google Scholar 

  25. Xie, Z.X., Li, T.Z., Ma, X., Wang, C.P.: Möbius geometry of three dimensional Wintgen ideal submanifolds In \({\mathbb{S}}^5\). Sci. China Math. 57, 1203–1220 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology (Beijing), Beijing, 100083, China

    Zhenxiao Xie

  2. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Tongzhu Li

  3. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Xiang Ma

  4. School of Mathematics and Computer Science and FJKLMAA, Fujian Normal University, Fuzhou, 350108, China

    Changping Wang

Authors
  1. Zhenxiao Xie
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  2. Tongzhu Li
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  3. Xiang Ma
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  4. Changping Wang
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Corresponding author

Correspondence to Xiang Ma.

Additional information

Supported by the Fundamental Research Funds for Central Universities. Supported by NSFC 11601513, 11571037, 11471021 and 11331002.

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Xie, Z., Li, T., Ma, X. et al. Wintgen ideal submanifolds: reduction theorems and a coarse classification. Ann Glob Anal Geom 53, 377–403 (2018). https://doi.org/10.1007/s10455-017-9581-1

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  • DOI: https://doi.org/10.1007/s10455-017-9581-1

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