Rigidity of Rank-One Factors of Compact Symmetric Spaces
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, p. 491-509
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.
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.
DOI : https://doi.org/10.5802/aif.2621
Classification:  53C40,  53C35,  53C42
Keywords: Minimal submanifolds, rigidity, symmetric spaces.
@article{AIF_2011__61_2_491_0,
     author = {Clarke, Andrew},
     title = {Rigidity of Rank-One Factors of Compact Symmetric Spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     pages = {491-509},
     doi = {10.5802/aif.2621},
     zbl = {1231.53044},
     mrnumber = {2895065},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2011__61_2_491_0}
}
Rigidity of Rank-One Factors of Compact Symmetric Spaces. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 491-509. doi : 10.5802/aif.2621. https://aif.centre-mersenne.org/item/AIF_2011__61_2_491_0/

[1] Barbosa, J.L.M. An extrinsic rigidity theorem for minimal immersions of S 2 into S n , J. Diff. Geom., Tome 14 (1979), pp. 355-368 | MR 594706 | Zbl 0427.53028

[2] Chavel, I. Riemannian Symmetric Spaces of Rank One, Marcel Dekker Inc., New York (1972) | MR 383304 | Zbl 0239.53032

[3] Chern, S.S.; Do Carmo, M.; Kobayashi, S. Minimal submanifolds of the sphere with second fundamental form of constant length, Springer Verlag, Functional Analysis and Related Fields (1970) | MR 273546 | Zbl 0216.44001

[4] Fischer-Colbrie, D. Some rigidity theorems for minimal submanifolds of the sphere, Acta Math., Tome 145 (1980), pp. 29-46 | Article | MR 558091 | Zbl 0464.53047

[5] Gluck, H.; Morgan, F.; Ziller, W. Calibrated geometries in Grassmann manifolds, Comment Math. Helv., Tome 64 (1989), pp. 256-268 | Article | MR 997365 | Zbl 0681.53039

[6] Hsiang, W.T.; Hsiang, W.Y. Examples of codimension-one closed minimal submanifolds in some symmetric spaces. I., J. Diff. Geom., Tome 15 (1980), pp. 543-551 | MR 628343 | Zbl 0467.53023

[7] Kobayashi, S.; Nomizu, K. Foundations of Differential Geometry, Interscience, New York Tome 2 (1969) | Zbl 0175.48504

[8] Lawson Jr., H.B. Complete minimal surfaces in S 3 , Ann. of Math., Tome 92 (1970), pp. 335-374 | Article | MR 270280 | Zbl 0205.52001

[9] Lawson Jr., H.B. Rigidity theorems in rank-1 symmetric spaces, J. Diff. Geom., Tome 4 (1970), pp. 349-357 | MR 267492 | Zbl 0199.56401

[10] Mok, N.; Siu, Y.T.; Yeung, S.-K. Geometric Superrigidity, J. Diff. Geom., Tome 113 (1993) no. 1, pp. 57-83 | MR 1223224 | Zbl 0808.53043

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

[12] Simons, J. Minimal varieties in riemannian manifolds, Ann. Math., Tome 88 (1968), pp. 65-105 | Article | MR 233295 | Zbl 0181.49702

[13] Thi, D.Č. Minimal real currents on compact riemannian manifolds, Math. USSR Izv., Tome 11 (1970), pp. 807-820 | Article | Zbl 0387.49042