Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-21T21:16:19.957Z Has data issue: false hasContentIssue false
  • English
  • Français

A general rigidity theorem for complete submanifolds

Published online by Cambridge University Press:  22 January 2016

Katsuhiro Shiohama  and
Hongwei Xu
Katsuhiro Shiohama
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan
Hongwei Xu
Affiliation:
Institute of mathematics, Zhejiang University, Hangzhou, 310027-China
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature normal in a complete and simply connected Riemannian (n+p) -manifold Nn+p with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number T(n,p) ∈ (0,1) with the property that if the sectional curvature of N is pinched in [T(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then Nn+p is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 150 , June 1998 , pp. 105 - 134
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

References

[AdC] H., Alencar and Carmo, M. do, Hypersurfaces with constant mean curvature in spheres, Proc. A. M. S., 120 (1994), 1223–1229.Google Scholar
[CDK] Chern, S. S., do Carmo, M. and Kobayashi, S., Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields (F. E. Browder, ed.), Springer-Verlag, New York (1970).Google Scholar
[Ga] Gauchman, H., Minimal submanifolds of a sphere with bounded second fundamental form, Trans. A. M. S., 298 (1986), 779–791.Google Scholar
[G] Goldberg, S., Curvature and Homology, Academic Press, London, 1962.Google Scholar
[H] Hasanis, T., Characterization of totally umbilical hypersurfaces, Proc. Amer. Math. Soc, 81 (1981), 447–450.Google Scholar
[L1] Lawson, B., Local rigidity theorems for minimal hypersurfaces, Ann. of Math., 89 (1969), 187–197.Google Scholar
[L2] Lawson, B., Rigidity theorems in rank one symmetric space, J. Differential Geometry, 4 (1970), 349–357.CrossRefGoogle Scholar
[LL] Li, A. M. and Li, J. M., An intrinsic rigidity theorem for minimal submanifolds in a sphere, Archiv der Math., 58 (1992), 582–594.Google Scholar
[O] Omori, H., Isometric immersion of Riemannian manifold, J. Math. Soc. Japan, 19 (1967), 205–214.Google Scholar
[P] Pan, Y. L., Pinching theorems of Simon’s type for complete minimal submanifolds in the sphere, Proc. Amer. Math. Soc, 93 (1985), 710–712.Google Scholar
[Sh] Shen, Y. B., On intrinsic rigidity for minimal submanifolds in a sphere, Sci. Sinica, Ser. A, 32 (1989), 769–781.Google Scholar
[Si] Simons, J., Minimal varieties in Riemannian manifolds, Ann. of Math., 88, (2) (1968), 62–105.Google Scholar
[X1] Xu, H. W., A rigidity theorem for submanifolds with parallel mean curvature in a sphere, Archiv der Math., 61 (1993), 489–496.Google Scholar
[X2] Xu, H. W., On closed minimal submanifolds in pinched Riemannian manifolds, Trans. A. M. S., 347 (1995), 1743–1751.Google Scholar
[X3] Xu, H. W., A pinching constant of Simons’ type and isometric immersion, Chinese Ann. Math., Ser. A, 12 (1991), 261–269.Google Scholar
[X4] Xu, H. W., Lower bounds for eigenvalues of submanifolds in a Riemannian manifold (1992), no. Research Report, Zhejiang University.Google Scholar
[Y1] Yau, S. T., Submanifolds with constant mean curvature I, II, Amer. J. Math., 96, 97 (1974, 1975), 346–366, 76–100.Google Scholar
[Y2] Yau, S. T., Harmonic functions on complete Riemannian manifolds, Comm. Pure and Appl. Math., 28 (1975), 201–228.Google Scholar