Glasgow Mathematical Journal
- Glasgow Mathematical Journal / Volume 39 / Issue 01 / January 1997, pp 29-33
- Copyright © Glasgow Mathematical Journal Trust 1997
- DOI: http://dx.doi.org/10.1017/S0017089500031876 (About DOI), Published online: 18 May 2009
Research Article
Normal curvature of minimal submanifolds in a sphere
Sharief Deshmukha1
a1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh-11451, Saudi Arabia
Simons [5] has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki [4] improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfies
then either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen [6] but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791).
(Received June 23 1995)
- 1. Barbosa, J. L. and do Carmo, M., Stability of minimal surfaces and eigenvalues of the Laplacian, Math. Z. 173 (1980), 13–28. [OpenURL Query Data] [CrossRef] [Google Scholar]
- 2. Cheng, S.-Y., Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), 289–297. [OpenURL Query Data] [CrossRef] [Google Scholar]
- 3. Chern, S. S., do Carmo, M. and Kobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, Functional analysis and related fields (Springer, 1970), 59–75. [Google Scholar]
- 4. Sakaki, M., Remarks on the rigidity and stability of minimal submanifolds, Proc. Amer. Math. Soc. 106 (1989), 793–795. [OpenURL Query Data] [CrossRef] [Google Scholar]
- 5. Simons, J., Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. [OpenURL Query Data] [CrossRef] [Google Scholar]
- 6. Shen, Y. B., Curvature pinching for the three-dimensional minimal submanifolds in a sphere, Proc. Amer. Math. Soc. 115 (1992), 791–795. [OpenURL Query Data] [CrossRef] [Google Scholar]
