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.
Keywordsminimal surface normal curvature Gauss curvature pinching
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.
- 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
- 15.Simon U. Eigenvalues of the Laplacian and minimal immersions into spheres. In: Differential Geometry. Montreal: Pitman, 1985, 115–120Google Scholar