no. 2
Rigidity of Rank-One Factors of Compact Symmetric Spaces
[Rigidité des facteurs de rang-1 des espaces symétriques compacts]
Clarke, Andrew1
1 Université de Nantes Laboratoire de Mathématiques Jean Leray 2, rue de la Houssinière - BP 92208 44322 Nantes Cedex 3 (France)
Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 491-509.

Nous considérons la décomposition d’un espace symétrique de type compact et nous montrons que les facteurs de rang 1, considérés comme sous-variétés de cet espace, sont isolés de toutes les sous-variétés minimales inéquivalentes.

We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds.

DOI : 10.5802/aif.2621
Classification : 53C40, 53C35, 53C42
Keywords: Minimal submanifolds, rigidity, symmetric spaces.
Mot clés : sous-varietés minimales, rigidité, espaces symétriques.
Clarke, Andrew 1

1 Université de Nantes Laboratoire de Mathématiques Jean Leray 2, rue de la Houssinière - BP 92208 44322 Nantes Cedex 3 (France) 
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     journal = {Annales de l'Institut Fourier},
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     volume = {61},
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 Clarke, Andrew. Rigidity of Rank-One Factors of Compact Symmetric Spaces. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 491-509. doi : 10.5802/aif.2621. http://www.numdam.org/articles/10.5802/aif.2621/

[1] Barbosa, J.L.M. An extrinsic rigidity theorem for minimal immersions of S 2 into S n , J. Diff. Geom., Volume 14 (1979), pp. 355-368 | MR | Zbl

[2] Chavel, I. Riemannian Symmetric Spaces of Rank One, Marcel Dekker Inc., New York, 1972 | MR | Zbl

[3] Chern, S.S.; Do Carmo, M.; Kobayashi, S. Minimal submanifolds of the sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer Verlag, 1970 | MR | Zbl

[4] Fischer-Colbrie, D. Some rigidity theorems for minimal submanifolds of the sphere, Acta Math., Volume 145 (1980), pp. 29-46 | DOI | MR | Zbl

[5] Gluck, H.; Morgan, F.; Ziller, W. Calibrated geometries in Grassmann manifolds, Comment Math. Helv., Volume 64 (1989), pp. 256-268 | DOI | MR | Zbl

[6] Hsiang, W.T.; Hsiang, W.Y. Examples of codimension-one closed minimal submanifolds in some symmetric spaces. I., J. Diff. Geom., Volume 15 (1980), pp. 543-551 | MR | Zbl

[7] Kobayashi, S.; Nomizu, K. Foundations of Differential Geometry, 2, Interscience, New York, 1969 | Zbl

[8] Lawson Jr., H.B. Complete minimal surfaces in S 3 , Ann. of Math., Volume 92 (1970), pp. 335-374 | DOI | MR | Zbl

[9] Lawson Jr., H.B. Rigidity theorems in rank-1 symmetric spaces, J. Diff. Geom., Volume 4 (1970), pp. 349-357 | MR | Zbl

[10] Mok, N.; Siu, Y.T.; Yeung, S.-K. Geometric Superrigidity, J. Diff. Geom., Volume 113 (1993) no. 1, pp. 57-83 | MR | Zbl

[11] Simon, L. Lecture Notes on Geometric Measure Theory, Australian National University, 1983

[12] Simons, J. Minimal varieties in riemannian manifolds, Ann. Math., Volume 88 (1968), pp. 65-105 | DOI | MR | Zbl

[13] Thi, D.Č. Minimal real currents on compact riemannian manifolds, Math. USSR Izv., Volume 11 (1970), pp. 807-820 | DOI | Zbl

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