Minimal surfaces in a unit sphere pinched by intrinsic curvature and normal curvature



We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere S n , under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property K + K N = 1 for the Gauss curvature K and the normal curvature K N if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively.


minimal surface normal curvature Gauss curvature pinching 


53C42 53A10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by Chern Institute of Mathematics. The author thanks the referees for their professional suggestions about this paper which led to various improvements.


  1. 1.
    Baker C, Nguyen H T. Codimension two surfaces pinched by normal curvature evolving by mean curvature flow. Ann Inst H Poincaré Anal Non Linéaire, 2017, 34: 1599–1610MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benko K, Kothe M, Semmler K D, et al. Eigenvalues of the Laplacian and curvature. Colloq Math, 1979, 42: 19–31MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Calabi E. Minimal immersions of surfaces in Euclidean spheres. J Differential Geom, 1967, 1: 111–125MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen B Y. Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann Global Anal Geom, 2010, 38: 145–160MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chern S S. On the minimal immersions of the two-sphere in a space of constant curvature. In: Problems in Analysis. A Symposium in Honor of Salomon Bochner (PMS-31). Princeton: Princeton University, 1969, 27–40Google Scholar
  6. 6.
    do Carmo M, Wallach N. Representations of compact groups and minimal immersions into spheres. J Differential Geom, 1970, 4: 91–104MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ge J Q, Tang Z Z. A proof of the DDVV conjecture and its equality case. Pacific J Math, 2008, 237: 87–95MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hou Z H, Yang D. Minimal surfaces in a unit sphere. J Math Res Appl, 2012, 32: 346–354.MathSciNetMATHGoogle Scholar
  9. 9.
    Kenmotsu K. On compact minimal surfaces with non-negative Gaussian curvature in a space of constant curvature: I. Tohoku Math J (2), 1973, 25: 469–479MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kenmotsu K. On compact minimal surfaces with non-negative Gaussian curvature in a space of constant curvature: II. Tohoku Math J (2), 1975, 27: 291–301MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kozlowski M, Simon U. Minimal immersion of 2-manifolds into spheres. Math Z, 1984, 186: 377–382MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lawson H B. Local rigidity theorems for minimal hypersurfaces. Ann of Math (2), 1969, 89: 187–197MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lu Z. Normal scalar curvature conjecture and its applications. J Funct Anal, 2011, 261: 1284–1308MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Peng C K, Tang Z Z. On surfaces immersed in Euclidean space R4. Sci China Math, 2010, 53: 251–256MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Simon U. Eigenvalues of the Laplacian and minimal immersions into spheres. In: Differential Geometry. Montreal: Pitman, 1985, 115–120Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsLiaoning UniversityShenyangChina

Personalised recommendations