Minimal submanifolds of a sphere with bounded second fundamental form
HTML articles powered by AMS MathViewer
- by Hillel Gauchman PDF
- Trans. Amer. Math. Soc. 298 (1986), 779-791 Request permission
Abstract:
Let $h$ be the second fundamental form of an $n$-dimensional minimal submanifold $M$ of a unit sphere ${S^{n + p}}(p \geqslant 2)$, $S$ be the square of the length of $h$, and $\sigma (u) = ||h(u,u)|{|^2}$ for any unit vector $u \in TM$. Simons proved that if $S \leqslant n/(2 - 1/p)$ on $M$, then either $S \equiv 0$, or $S \equiv n/(2 - 1/p)$. Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying $S \equiv n/(2 - 1/p)$. In this paper the analogous results for $\sigma (u)$ are obtained. It is proved that if $\sigma (u) \leqslant \tfrac {1} {3}$, then either $\sigma (u) \equiv 0$, or $\sigma (u) \equiv \tfrac {1} {3}$. All minimal submanifolds satisfying $\sigma (u)$ are determined. A stronger result is obtained if $M$ is odd-dimensional.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
- H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
- Ngaiming Mok and Jia Qing Zhong, Curvature characterization of compact Hermitian symmetric spaces, J. Differential Geom. 23 (1986), no. 1, 15–67. MR 840400
- Barrett O’Neill, Isotropic and Kähler immersions, Canadian J. Math. 17 (1965), 907–915. MR 184181, DOI 10.4153/CJM-1965-086-7
- Kunio Sakamoto, Planar geodesic immersions, Tohoku Math. J. (2) 29 (1977), no. 1, 25–56. MR 470913, DOI 10.2748/tmj/1178240693
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 298 (1986), 779-791
- MSC: Primary 53C42
- DOI: https://doi.org/10.1090/S0002-9947-1986-0860393-5
- MathSciNet review: 860393