Abstract
LetM be ann-dimensional compact minimal submanifold of a unit sphereS n+p (p≥2); and letS be a square of the length of the second fundamental form. IfS≤2/3n everywhere onM, thenM must be totally geodesic or a Veronese surface.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Simons, J., ‘Minimal varieties in Riemannian manifolds’,Ann. Math. 88 (1968), 62–105.
Chern, S. S., Do Carmo, M. and Kobayashi, S., ‘Minimal submanifolds of sphere with second fundamental form of constant length’,Shiing-Shen Chern Selected Papers, Springer-Verlag 1978, pp. 393–409.
Yau, S. T., ‘Submanifolds with constant mean curvature (II)’,Amer. J. Math. 97 (1975), 76–100.
Itoh, T., Addendum to paper ‘On Veronese manifolds’,J. Math. Soc. Japan 30 (1978), 73–74.
Shen, Yibing, ‘On intrinsic rigidity for minimal submanifolds in a sphere’ (in Chinese),Sci. China. Ser. A 32, No. 7 (1989), 769–781.
Cauchman, H., ‘Minimal submanifold of a sphere with bounded second fundamental form’,Trans. Amer. Math. Soc. 298 (1986), 779–791.
Benko, K., Kothe, M., Semmler, K.-D. and Simon, U., ‘Eigenvalues of Laplacian and curvature’,Colloq. Math. 42 (1979), 19–31.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, Q., Xu, S. Rigidity of compact minimal submanifolds in a unit sphere. Geom Dedicata 45, 83–88 (1993). https://doi.org/10.1007/BF01667404
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01667404