Ln/2-pinching theorems for submanifolds with parallel mean curvature in a sphere

Hong-Wei XU

doi:10.2969/jmsj/04630503

L_n/2-pinching theorems for submanifolds with parallel mean curvature in a sphere

Hong-Wei XU

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JOURNAL FREE ACCESS

1994 Volume 46 Issue 3 Pages 503-515

DOI https://doi.org/10.2969/jmsj/04630503

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Published: 1994 Received: October 12, 1992 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : L_n/2-pinching theorems for submanifolds with parallel mean curvature in a sphere
Right : L_n/2-pinching theorems for submanifolds with parallel mean curvature in a sphere

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom., 1 (1967), 111-125.
2) B.Y. Chen, Geometry of Submanifolds, Marcel Dekker Inc., New York, 1973.
3) X.P. Chen, Harmonic mapping and Gauss mapping, Chinese Ann. Math., 4A(1983), 449-456.
4) S.S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional analysis and related fields, (ed. F.E. Browder), Springer, 1970.
5) D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715-727; 28 (1975), 765-766.
6) J.M. Lin and C.Y. Xia, Global pinching theorems for even dimensional minimal submanifolds in a unit sphere, Math. Z., 201 (1989), 381-389.
7) C.K. Peng and C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, in Seminar of minimal submanifolds, (ed. E. Bombieri), Princeton University Press, 1983.
8) R. Schoen and S.T. Yau, Differential Geometry, Academic Press, 1988.
9) C.L. Shen, A global pinching theorem for minimal hypersurfaces in a sphere, Proc. Amer. Math. Soc., 105 (1989), 192-198.
10) J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.
11) H.W. Xu, Some results on geometry of Riemannian submanifolds, Ph. D. dissertation, Fudan University, 1990.
12) S.T. Yau, Submanifolds with constant mean curvature I, II, Amer. J. Math., 96 (1974), 346-366; 97 (1975), 76-100.

Right : [1] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom., 1 (1967), 111-125.
[2] B. Y. Chen, Geometry of Submanifolds, Marcel Dekker Inc., New York, 1973.
[3] X. P. Chen, Harmonic mapping and Gauss mapping, Chinese Ann. Math., 4A (1983), 449-456.
[4] S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional analysis and related fields, (ed. F. E. Browder), Springer, 1970.
[5] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715-727; 28 (1975), 765-766.
[6] J. M. Lin and C. Y. Xia, Global pinching theorems for even dimensional minimal submanifolds in a unit sphere, Math. Z., 201 (1989), 381-389.
[7] C. K. Peng and C. L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, in Seminar of minimal submanifolds, (ed. E. Bombieri), Princeton University Press, 1983.
[8] R. Schoen and S. T. Yau, Differential Geometry, Academic Press, 1988.
[9] C. L. Shen, A global pinching theorem for minimal hypersurfaces in a sphere, Proc. Amer. Math. Soc., 105 (1989), 192-198.
[10] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.
[11] H. W. Xu, Some results on geometry of Riemannian submanifolds, Ph. D. dissertation, Fudan University, 1990.
[12] S. T. Yau, Submanifolds with constant mean curvature I, II, Amer. J. Math., 96 (1974), 346-366; 97 (1975), 76-100.

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