Abstract
In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS n+1(1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n 2(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cheng, Q. M.,The classification of complete hypersurfaces with constant mean curvature of space formof dimension 4, Mem. Fac. Sci. Kyushu Univ.47 (1993), 79–102.
Chern, S. S., Do Carmo, M. andKobayashi, S.,Minimal submanifolds of a sphere with second fundamental form of constant length, Functional analysis and related fields, edited by F. Browder, Springer-Verlag, Berlin 1970, 59–75.
Lawson, H. B.,Local rigidity theorems for minimal hypersurfaces, Ann. of Math.89 (1969), 179–185.
Otsuki, T.,Minimal hypersurfaces ina Riemannian manifold of constant curvature, Amer. J. Math.92 (1970), 145–173.
Yang, H. C. andCheng, Q. M.,A note on the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Chinese Science Bull.36 (1991), 1–6.
Yang, H. C. andCheng, Q. M.,An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit spheres, Manuscripta Math.82 (1994), 89–100.
Author information
Authors and Affiliations
Additional information
The Project Supported by NNSFC, CPSF and FECC.
Rights and permissions
About this article
Cite this article
Cheng, QM. The rigidity of Clifford torus\(S^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)\) . Commentarii Mathematici Helvetici 71, 60–69 (1996). https://doi.org/10.1007/BF02566409
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02566409