Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Minimal submanifolds of a sphere with bounded second fundamental form


Author: Hillel Gauchman
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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) = \vert\vert h(u,u)\vert{\vert^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 [Enhancements On Off] (What's this?)

  • [1] S.-S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer-Verlag, Berlin and New York, 1970, pp. 59-75. MR 0273546 (42:8424)
  • [2] B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187-197. MR 0238229 (38:6505)
  • [3] N. Mok and J.-Q. Zhang, Curvature characterization of compact Hermitian symmetric spaces, Trans. Amer. Math. Soc. 23 (1986), 15-67. MR 840400 (87k:53161)
  • [4] B. O'Neill, Isotropic and Kähler immersions, Canad. J. Math. 17 (1965), 907-915. MR 0184181 (32:1654)
  • [5] K. Sakamoto, Planar geodesic immersions, Tôhoku Math. J. 29 (1977), 25-56. MR 0470913 (57:10657)
  • [6] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. MR 0233295 (38:1617)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C42

Retrieve articles in all journals with MSC: 53C42


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0860393-5
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society