Skip to main content

Rigidity Results for Self-Shrinking Surfaces in ℝ4

Abstract

In this paper, we give some rigidity results for complete self-shrinking surfaces properly immersed in ℝ4 under some assumptions regarding their Gauss images. More precisely, we prove that this has to be a plane, provided that the images of either Gauss map projection lies in an open hemisphere or \({{\mathbb{S}}^2}(1/\sqrt 2 )\backslash \bar {\mathbb{S}}_ + ^1(1/\sqrt 2 )\). We also give the classification of complete self-shrinking surfaces properly immersed in ℝ4 provided that the images of Gauss map projection lies in some closed hemispheres. As an application of the above results, we give a new proof for the result of Zhou. Moreover, we establish a Bernstein-type theorem.

This is a preview of subscription content, access via your institution.

References

  1. Osserman R. Proof of a conjecture of Nirenberg. Comm Pure Appl Math, 1959, 12: 229–232

    MathSciNet  Article  Google Scholar 

  2. Fujimoto H. On the number of exceptional values of the Gauss map of minimal surfaces. J Math Soc Japan, 1988, 40: 235–247

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  5. Assimos R, Jost J. The geometry of maximum principles and a Bernstein theorem in codimension 2. arXiv:1811.09869, 2019

  6. Jost J, Xin Y L, Yang L. The regularity of harmonic maps into spheres and applications to Bernstein problems. J Differ Geom, 2012, 90: 131–176

    MathSciNet  Article  Google Scholar 

  7. Jost J, Xin Y L, Yang L. The Gauss image of entire graphs of higher codimension and Bernstein type theorems. Calc Var Partial Differential Equations, 2013, 47: 711–737

    MathSciNet  Article  Google Scholar 

  8. Bernstein S. Sur les surfaces définies au moyen de leur courbure moyenne ou totale. Ann Ec Norm Sup, 1910, 27: 233–256

    Article  Google Scholar 

  9. Chern S S. On the curvature of a piece of hypersurface in Euclidean space. Abh Math Sem Hamburg, 1965, 29: 77–91

    MathSciNet  Article  Google Scholar 

  10. 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, 1982, 57: 519–531

    MathSciNet  Article  Google Scholar 

  11. Jiang X Y, Sun H J, Zhao P B. Rigidity and gap results for the morse index of self-Shrinkers with any codimension. Results Math, 2019, 74: 68

    MathSciNet  Article  Google Scholar 

  12. Cheng Q M, Hori H, Wei G. Complete Lagrangian self-shrinkers in ℝ4. arXiv:1802.02396, 2018

  13. Wang L. A Bernstein type theorem for self-similar shrinkers. Geom Dedicata, 2011, 151: 297–303

    MathSciNet  Article  Google Scholar 

  14. Ding Q, Xin Y L, Yang L. The rigidity theorems of self-shrinkers via Gauss maps. Adv Math, 2016, 303: 151–174

    MathSciNet  Article  Google Scholar 

  15. Zhou H. A Bernstein type result for graphical self-shrinkers in ℝ4. Int Math Res Not, 2018, 21: 6798–6815

    MathSciNet  Article  Google Scholar 

  16. Abresch U, Langer J. The normalized curve shortening flow and homothetic solutions. J Differ Geom, 1986, 23(2): 175–196

    MathSciNet  Article  Google Scholar 

  17. Basto-Gonçalves J. The Gauss map for Lagrangean and isoclinic surfaces. arxiv:1304.2237, 2013

  18. Li H, Wang X. New characterizations of the Clifford torus as a Lagrangian self-shrinker. J Geom Anal, 2017, 27: 1393–1412

    MathSciNet  Article  Google Scholar 

  19. Li X X, Li X. On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane ℂ2. Acta Math Sci, 2019, 39B(6): 1695–1712

    Article  Google Scholar 

  20. Little J. On singularities of submanifolds of higher dimensional Euclidean spaces. Ann Mat Pura ed Appl, 1969, 83: 261–335

    MathSciNet  Article  Google Scholar 

  21. Borisenko A A, Nikolaevskil Y A. Grassman manifolds and the Grassmann image of submanifolds. Usp Mat Nauk, 1991, 46(2): 41–83

    Google Scholar 

  22. Lichnerowicz A. Applications harmoniques et variétés kähleriennes//Symposia Mathematica. London: Academic Press, 1969: 341–402

    Google Scholar 

  23. Course N. f-Harmonic Maps [D]. Warwick: University of Warwick, 2004

    MATH  Google Scholar 

  24. Rimoldi M, Veronelli G. Topology of steady and expanding gradient Ricci solitons via f-harmonic maps. Differ Geom Appl, 2013, 31(5): 623–638

    MathSciNet  Article  Google Scholar 

  25. Hoffman D A, Osserman R. The Gauss map of surfaces in ℝn. J Differ Geom, 1983, 18: 733–754

    Article  Google Scholar 

  26. Hoffman D A, Osserman R. The Gauss map of surfaces in ℝ3 and ℝ4. Proc London Math Soc, 1985, 50(3): 27–56

    MathSciNet  Article  Google Scholar 

  27. Cheng X, Zhou D. Volume estimates about shrinkers. Proc Amer Math Soc, 2013, 141: 687–696

    MathSciNet  Article  Google Scholar 

  28. Smoczyk K. Self-shrinkers of the mean curvature flow in arbitrary codimension. Int Math Res Not, 2005, 48: 2983–3004

    MathSciNet  Article  Google Scholar 

  29. Colding T H, Minicozzi II W P. Generic mean curvature flow I: generic singularities. Ann Math, 2012, 175: 755–833

    MathSciNet  Article  Google Scholar 

  30. Enomoto K. The Gauss image of flat surfaces in ℝ4. Kodai Math J, 1986, 9: 19–32

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Xuyong Jiang or Hejun Sun.

Additional information

This work was supported by the National Natural Science Foundation of China (11001130, 11871275) and the Fundamental Research Funds for the Central Universities (30917011335).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jiang, X., Sun, H. & Zhao, P. Rigidity Results for Self-Shrinking Surfaces in ℝ4. Acta Math Sci 41, 1417–1427 (2021). https://doi.org/10.1007/s10473-021-0502-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-021-0502-9

Key words

  • self-shrinkers
  • Gauss map
  • Bernstein-type theorem
  • rigidity

2010 MR Subject Classification

  • 53C24
  • 53C42