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On the Chern conjecture for isoparametric hypersurfaces

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Abstract

For a closed hypersurface Mn ⊂ Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, we show that if \({\rm{tr}}\left({{{\cal A}^k}} \right)\) are constants for k = 3, …, n − 1 and the shape operator \({\cal A}\) then M is isoparametric. The result generalizes the theorem of de Almeida and Brito (1990) for n = 3 to any dimension n, strongly supporting the Chern conjecture.

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References

  1. Cartan E. Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math Z, 1939, 45: 335–367

    Article  MathSciNet  MATH  Google Scholar 

  2. Cecil T E, Chi Q S, Jensen G R. Isoparametric hypersurfaces with four principal curvatures. Ann of Math (2), 2007, 166: 1–76

    Article  MathSciNet  MATH  Google Scholar 

  3. Cecil T E, Ryan P J. Geometry of Hypersurfaces. Springer Monographs in Mathematics. New York: Springer, 2015

    Book  MATH  Google Scholar 

  4. Chang S P. A closed hypersurface with constant scalar curvature and constant mean curvature in \({\mathbb{S}^4}\) is isoparametric. Comm Anal Geom, 1993, 1: 71–100

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang S P. On minimal hypersurfaces with constant scalar curvatures in S4. J Differential Geom, 1993, 37: 523–534

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng Q M, Wan Q R. Hypersurfaces of space forms M4(c) with constant mean curvature. In: Geometry and Global Analysis. Sendai: Tohoku Univ, 1993, 437–442

    Google Scholar 

  7. Chern S S. Minimal Submanifolds in a Riemannian Manifold. Lawrence: University of Kansas, 1968

    Google Scholar 

  8. Chern S S, do Carmo M, Kobayashi S. Minimal submanifolds of a sphere with second fundamental form of constant length. In: Functional Analysis and Related Fields. Berlin: Springer, 1970, 59–75

    Google Scholar 

  9. Chi Q S. Isoparametric hypersurfaces with four principal curvatures, II. Nagoya Math J, 2011, 204: 1–18

    Article  MathSciNet  MATH  Google Scholar 

  10. Chi Q S. Isoparametric hypersurfaces with four principal curvatures, III. J Differential Geom, 2013, 94: 469–504

    Article  MathSciNet  MATH  Google Scholar 

  11. Chi Q S. Isoparametric hypersurfaces with four principal curvatures, IV. J Differential Geom, 2020, 115: 225–301

    Article  MathSciNet  MATH  Google Scholar 

  12. de Almeida S C, Brito F G B. Closed 3-dimensional hypersurfaces with constant mean curvature and constant scalar curvature. Duke Math J, 1990, 61: 195–206

    Article  MathSciNet  MATH  Google Scholar 

  13. Deng Q T, Gu H L, Wei Q Y. Closed Willmore minimal hypersurfaces with constant scalar curvature in \({\mathbb{S}^5}\left(1 \right)\) are isoparametric. Adv Math, 2017, 314: 278–305

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding Q, Xin Y L. On Chern’s problem for rigidity of minimal hypersurfaces in the spheres. Adv Math, 2011, 227: 131–145

    Article  MathSciNet  MATH  Google Scholar 

  15. Dorfmeister J, Neher E. Isoparametric hypersurfaces, case g = 6, m = 1. Comm Algebra, 1985, 13: 2299–2368

    Article  MathSciNet  MATH  Google Scholar 

  16. Ge J Q, Tang Z Z. Chern conjecture and isoparametric hypersurfaces. In: Differential Geometry: Under the Influence of S. S. Chern. Beijing-Boston: Higher Education Press and International Press, 2012, 49–60

    Google Scholar 

  17. Immervoll S. On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann of Math (2), 2008, 168: 1011–1024

    Article  MathSciNet  MATH  Google Scholar 

  18. Lawson H B Jr. Local rigidity theorems for minimal hypersurfaces. Ann of Math (2), 1969, 89: 167–179

    Article  MathSciNet  MATH  Google Scholar 

  19. Lei L, Xu H W, Xu Z Y. On Chern’s conjecture for minimal hypersurfaces in spheres. arXiv:1712.01175, 2017

  20. Lusala T, Scherfner M, Sousa L A M Jr. Closed minimal Willmore hypersurfaces of \({\mathbb{S}^5}\left(1 \right)\) with constant scalar curvature. Asian J Math, 2005, 9: 65–78

    Article  MathSciNet  MATH  Google Scholar 

  21. Milnor J. Hyperbolic geometry: The first 150 years. Bull Amer Math Soc (NS), 1982, 6: 9–24

    Article  MathSciNet  MATH  Google Scholar 

  22. Miyaoka R. Isoparametric hypersurfaces with (g, m) = (6, 2). Ann of Math (2), 2013, 177: 53–110

    Article  MathSciNet  MATH  Google Scholar 

  23. Miyaoka R. Errata of “isoparametric hypersurfaces with (g, m) = (6, 2)”. Ann of Math (2), 2016, 183: 1057–1071

    Article  MathSciNet  MATH  Google Scholar 

  24. Münzner H F. Isoparametrische Hyperflächen in Sphären, I. Math Ann, 1980, 251: 57–71

    Article  MathSciNet  MATH  Google Scholar 

  25. Münzner H F. Isoparametrische Hyperflächen in Sphären, II. Math Ann, 1981, 256: 215–232

    Article  MathSciNet  MATH  Google Scholar 

  26. Peng C K, Terng C L. Minimal hypersurfaces of spheres with constant scalar curvature. Ann of Math Stud, 1983, 103: 177–198

    MathSciNet  MATH  Google Scholar 

  27. Peng C K, Terng C L. The scalar curvature of minimal hypersurfaces in spheres. Math Ann, 1983, 266: 105–113

    Article  MathSciNet  MATH  Google Scholar 

  28. Scherfner M, Vrancken L, Weiss S. On closed minimal hypersurfaces with constant scalar curvature in \({\mathbb{S}^7}\). Geom Dedicata, 2012, 161: 409–416

    Article  MathSciNet  MATH  Google Scholar 

  29. Scherfner M, Weiss S, Yau S T. A review of the Chern conjecture for isoparametric hypersurfaces in spheres. In: Advances in Geometric Analysis. Advanced Lectures in Mathematics (ALM), vol. 21. Somerville: Int Press, 2012, 175–187

  30. Simons J. Minimal varieties in Riemannian manifolds. Ann of Math (2), 1968, 88: 62–105

    Article  MathSciNet  MATH  Google Scholar 

  31. Suh Y J, Yang H Y. The scalar curvature of minimal hypersurfaces in a unit sphere. Commun Contemp Math, 2007, 9: 183–200

    Article  MathSciNet  MATH  Google Scholar 

  32. Tang Z Z, Wei D Y, Yan W J. A sufficient condition for a hypersurface to be isoparametric. Tohoku Math J (2), 2020, 72: 493–505

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang Z Z, Yan W J. Isoparametric foliation and Yau conjecture on the first eigenvalue. J Differential Geom, 2013, 94: 521–540

    Article  MathSciNet  MATH  Google Scholar 

  34. Tang Z Z, Yan W J. Isoparametric foliation and a problem of Besse on generalizations of Einstein condition. Adv Math, 2015, 285: 1970–2000

    Article  MathSciNet  MATH  Google Scholar 

  35. Verstraelen L. Sectional curvature of minimal submanifolds. In: Proceedings of Workshop on Differential Geometry. Southampton: University of Southampton, 1986, 48–62

    Google Scholar 

  36. Yang H C, Cheng Q M. Chern’s conjecture on minimal hypersurfaces. Math Z, 1998, 227: 377–390

    Article  MathSciNet  MATH  Google Scholar 

  37. Yau S-T. Problem section. In: Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102. Princeton: Princeton University Press, 1982, 669–706

    Google Scholar 

  38. Yau S-T. Selected Expository Works of Shing-Tung Yau with Commentary, Volume 1. Advanced Lectures in Mathematics Series, vol. 28. Somerville: Int Press, 2014

    MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11722101, 11871282 and 11931007), Beijing Natural Science Foundation (Grant No. Z190003) and Nankai Zhide Foundation.

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Authors and Affiliations

  1. Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China

    Zizhou Tang

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China

    Wenjiao Yan

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  1. Zizhou Tang
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  2. Wenjiao Yan
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Correspondence to Wenjiao Yan.

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Tang, Z., Yan, W. On the Chern conjecture for isoparametric hypersurfaces. Sci. China Math. 66, 143–162 (2023). https://doi.org/10.1007/s11425-022-1967-4

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  • DOI: https://doi.org/10.1007/s11425-022-1967-4

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