A global pinching theorem for compact minimal surfaces in $S^ 3$
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- by Yi-Jung Hsu PDF
- Proc. Amer. Math. Soc. 113 (1991), 1041-1044 Request permission
Abstract:
Let $M$ be a compact minimally immersed surface in the unit sphere ${S^3}$, and let $S$ denote the square of the length of the second fundamental form of $M$. We prove that if ${\left \| S \right \|_2} \leq 2\sqrt 2 \pi$, then $M$ is either the equatorial sphere or the Clifford torus.References
- S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR 0273546
- Shiing-shen Chern and Richard K. Lashof, On the total curvature of immersed manifolds. II, Michigan Math. J. 5 (1958), 5–12. MR 97834
- H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
- Chun Li Shen, A global pinching theorem of minimal hypersurfaces in the sphere, Proc. Amer. Math. Soc. 105 (1989), no. 1, 192–198. MR 973845, DOI 10.1090/S0002-9939-1989-0973845-2
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 1041-1044
- MSC: Primary 53C42; Secondary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1086331-8
- MathSciNet review: 1086331