A sufficient condition for a hypersurface to be isoparametric. (English) Zbl 1479.53047
Summary: Let \(M^n\) be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose \(\mathfrak{a}\) is a symmetric \((0,2)\) tensor field whose dual \((1,1)\) tensor \(\mathcal{A}\) has \(n\) distinct eigenvalues, and \(\mathrm{tr} (\mathcal{A}^k)\) are constants for \(k=1,\ldots, n-1\). We show that all the eigenvalues of \(\mathcal{A}\) are constants, generalizing a theorem of S. C. de Almeida and F. G. B. Brito [Duke Math. J. 61, No. 1, 195–206 (1990; Zbl 0721.53056)] to higher dimensions.
As a consequence, a closed hypersurface \(M^n\) in \(S^{n+1}\) is isoparametric if one takes \(\mathfrak{a}\) above to be the second fundamental form, giving affirmative evidence to Chern’s conjecture.
As a consequence, a closed hypersurface \(M^n\) in \(S^{n+1}\) is isoparametric if one takes \(\mathfrak{a}\) above to be the second fundamental form, giving affirmative evidence to Chern’s conjecture.
Citations:
Zbl 0721.53056References:
| [1] | U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures, Math. Ann. 264 (1983), 283-302. · Zbl 0505.53027 · doi:10.1007/BF01459125 |
| [2] | E. Cartan, Sur des familles d’hypersurfaces isoparamétriques des espaces sphériques á 5 et á 9 dimensions, Univ. Nac. Tucumán. Revista A. 1 (1940), 5-22. · Zbl 0025.22603 |
| [3] | T. E. Cecil, Q. S. Chi and G. R. Jensen, Isoparametric hypersurfaces with four principal curvatures, Ann. Math. 166 (2007), no. 1, 1-76. · Zbl 1143.53058 · doi:10.4007/annals.2007.166.1 |
| [4] | S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of the sphere with second fundamental form of constant length, in: F. Browder (Ed.), Functional Analysis and Related Fields, Springer-Verlag, Berlin, 1970. · Zbl 0216.44001 |
| [5] | S. P. Chang, On minimal hypersurfaces with constant scalar curvatures in \(S^4\), J. Differential Geom. 37 (1993), 523-534. · Zbl 0838.53046 |
| [6] | S. P. Chang, A closed hypersurface with constant scalar curvature and constant mean curvature in \(S^4\) is isoparametric, Comm. Anal. Geom. 1 (1993), 71-100. · Zbl 0791.53058 |
| [7] | S. S. Chern, Minimal submanifolds in a Riemannian manifold, Mimeographed Lecture Note, Univ. of Kansas, 1968. |
| [8] | Q. S. Chi, Isoparametric hypersurfaces with four principal curvatures, II, Nagoya Math. J. 204 (2011), 1-18. · Zbl 1243.53094 · doi:10.1215/00277630-1431813 |
| [9] | Q. S. Chi, Isoparametric hypersurfaces with four principal curvatures, III, J. Differential Geom. 94 (2013), 469-504. · Zbl 1280.53053 · doi:10.4310/jdg/1370979335 |
| [10] | Q. S. Chi, Isoparametric hypersurfaces with four principal curvatures, IV, arXiv: 1605.00976, 2016. · Zbl 1455.53081 · doi:10.4310/jdg/1589853626 |
| [11] | S. C. de Almeida and F. G. B. Brito, Closed 3-dimensional hypersurfaces with constant mean curvature and constant scalar curvature, Duke Math. J. 61 (1990), 195-206. · Zbl 0721.53056 · doi:10.1215/S0012-7094-90-06109-5 |
| [12] | J. Dorfmeister and E. Neher, Isoparametric hypersurfaces, case \(g = 6, m =1\), Comm. Algebra 13 (1985), 2299-2368. · Zbl 0578.53041 · doi:10.1080/00927878508823278 |
| [13] | F. Q. Fang, On the topology of isoparametric hypersurfaces with four distinct principal curvatures, Proc. Amer. Math. Soc. 127 (1999), 259-264. · Zbl 0914.53033 |
| [14] | J. Q. Ge and Z. Z. Tang, Isoparametric functions and exotic spheres, J. Reine Angew. Math. 683 (2013), 161-180. · Zbl 1279.53057 |
| [15] | S. Immervoll, On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. of Math. 168 (2008), 1011-1024. · Zbl 1176.53057 |
| [16] | H. B. Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. Math. 89 (1969), 167-179. |
| [17] | R. Miyaoka, Isoparametric hypersurfaces with \((g, m) = (6, 2)\), Ann. of Math. 177 (2013), 53-110. · Zbl 1263.53049 |
| [18] | K. Nomizu, Characteristic roots and vectors of a differentiable family of symmetric matrices, Linear and Multilinear Algebra 1 (1973), no. 2, 159-162. · Zbl 0277.15004 · doi:10.1080/03081087308817014 |
| [19] | C. K. Peng and C. L. Terng, Minimal hypersurfaces of spheres with constant scalar curvature, Seminar on Minimal Submanifolds, Ann. Math. Stud., Princeton Univ. Press, Princeton, NJ, 1983, 177-198. · Zbl 0534.53048 |
| [20] | C. Qian and Z. Z. Tang, Isoparametric functions on exotic spheres, Adv. Math. 272 (2015), 611-629. · Zbl 1312.53059 · doi:10.1016/j.aim.2014.12.020 |
| [21] | J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62-105. · Zbl 0181.49702 · doi:10.2307/1970556 |
| [22] | S. Stolz, Multiplicities of Dupin hypersurfaces, Invent. Math. 138 (1999), 253-279. · Zbl 0944.53035 · doi:10.1007/s002220050378 |
| [23] | Y. J. Suh and H. Y. Yang, The scalar curvature of minimal hypersurfaces in a unit sphere, Commun. Contemp. Math. 9 (2007), 183-200. · Zbl 1123.53031 |
| [24] | Z. Z. Tang, Isoparametric hypersurfaces with four distinct principal curvatures, Chinese Sci. Bull. 36 (1991), 1237-1240. · Zbl 0746.53047 · doi:10.1360/csb1991-36-16-1237 |
| [25] | Z. Z. Tang, Y. Q. Xie and W. J. Yan, Schoen-Yau-Gromov-Lawson theory and isoparametric foliation, Comm. Anal. Geom. 20 (2012), 989-1018. · Zbl 1272.53027 |
| [26] | Z. Z. Tang, Y. Q. Xie and W. J. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, II, J. Funct. Anal. 266 (2014), 6174-6199. · Zbl 1300.53062 · doi:10.1016/j.jfa.2014.02.024 |
| [27] | Z. Z. Tang and W. J. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, J. Differential Geom. 94 (2013), 521-540. · Zbl 1277.53058 · doi:10.4310/jdg/1370979337 |
| [28] | Z. Z. Tang and W. J. Yan, Isoparametric foliation and a problem of Besse on generalizations of Eisntein condition, Adv. Math. 285 (2015), 1970-2000. · Zbl 1327.53060 |
| [29] | H. C. Yang and Q. M. Cheng, Chern’s conjecture on minimal hypersurfaces, Math. Z. 227 (1998), 377-390. · Zbl 0893.53025 · doi:10.1007/PL00004382 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
