Recent Progress in Isoparametric Functions and Isoparametric Hypersurfaces

  • Chao Qian
  • Zizhou Tang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions.
  1. (1)

    Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered.

  2. (2)

    The Yau conjecture on the first eigenvalues of the embedded minimal hypersurfaces in the unit spheres. The history and progress of the Yau conjecture on minimal isoparametric hypersurfaces will be stated.



Riemannian Manifold Unit Sphere Complete Riemannian Manifold Positive Scalar Curvature Isoparametric Hypersurface 
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  1. 1.
    Bérard-Bergery, L.: Quelques exemples de variétés riemanniennes où toutes les géodésiques issues dun point sont fermées et de même longueur suivis de quelques résultats sur leur topologie. Ann. Inst. Fourier 27, 231–249 (1977)CrossRefMATHGoogle Scholar
  2. 2.
    Besse, A.L.: Manifolds all of whose geodesics are closed, with appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger, and J. L. Kazdan. Ergeb. Math. Grenzgeb., vol. 93. Springer, Berlin (1978)Google Scholar
  3. 3.
    Boyer, C.P., Galicki, K., Kollár, J.: Einstein metrics on spheres. Ann. Math. 162, 557–580 (2005)CrossRefMATHGoogle Scholar
  4. 4.
    Brendle, S.: Embedded minimal tori in S 3 and the Lawson conjecture. Acta Math. 11, 177–190 (2013)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cartan, E.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. Mat. 17, 177–191 (1938)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cartan, E.: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45, 335–367 (1939)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cecil, T.E.: Isoparametric and Dupin hypersurfaces. SIGMA 4, Paper 062, 28 pp. (2008)Google Scholar
  8. 8.
    Cecil, T.E., Ryan, P.T.: Tight and Taut Immersions of Manifolds. Research Notes in Mathematics, vol. 107. Pitman, London (1985)Google Scholar
  9. 9.
    Cecil, T.E., Chi, Q.S., Jensen, G.R.: Isoparametric hypersurfaces with four principal curvatures. Ann. Math. 166(1), 1–76 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Chi, Q.S.: The isoparametric story. National Taiwan University, 25 June–6 July 2012.
  11. 11.
    Chi, Q.S.: Isoparametric hypersurfaces with four principal curvatures, III. J. Differ. Geom. 94, 487–504 (2013)Google Scholar
  12. 12.
    Choi, H.I., Wang, A.N.: A first eigenvalue estimate for minimal hypersurfaces. J. Differ. Geom. 18, 559–562 (1983)MATHMathSciNetGoogle Scholar
  13. 13.
    Dorfmeister, J., Neher, E.: Isoparametric hypersurfaces, case g = 6, m = 1. Commun. Algebra 13, 2299–2368 (1985)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Ferus, D., Karcher, H., Münzner, H.F.: Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z. 177, 479–502 (1981). For an English version, see arXiv: 1112.2780Google Scholar
  15. 15.
    Freedman, M.: The topology of four dimensional manifolds. J. Differ. Geom. 17(3), 357–453 (1982)MATHGoogle Scholar
  16. 16.
    Ge, J.Q., Tang, Z.Z.: Chern conjecture and isoparametric hypersurfaces. In: Shen, Y.B., Shen, Z.M., Yau, S.T. (eds.) Differential Geometry–under the Influence of S.S.Chern. Higher Education Press and International Press, Beijing, Boston (2012)Google Scholar
  17. 17.
    Ge, J.Q., Tang, Z.Z.: Isoparametric functions and exotic spheres. J. Reine Angew. Math. 683, 161–180 (2013)MATHMathSciNetGoogle Scholar
  18. 18.
    Ge, J.Q., Tang, Z.Z.: Geometry of isoparametric hypersurfaces in Riemannian manifolds. Asian J. Math. 18, 117–126 (2014)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ge, J.Q., Tang, Z.Z., Yan, W.J.: A filtration for isoparametric hypersurfaces in Riemannian manifolds. J. Math. Soc. Jpn. (2013) (to appear)Google Scholar
  20. 20.
    Gromoll, D., Meyer, W.: An exotic sphere with nonnegative sectional curvature. Ann. Math. 100, 401–406 (1974)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. 152, 331–367 (2000)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hsiang, W.C., Hsiang, W.Y.: On compact subgroups of the diffeomorphism groups of Kervaire spheres. Ann. Math. 85, 359–369 (1967)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Immervoll, S.: On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann. Math. 168(3), 1011–1024 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Joachim, M., Wraith, D.J.: Exotic spheres and curvature. Bull. Am. Math. Soc. (N.S.) 45(4), 595–616 (2008)Google Scholar
  25. 25.
    Kervaire, M., Milnor, J.: Groups of homotopy spheres. I. Ann. Math. (2) 77, 504–537 (1963)Google Scholar
  26. 26.
    Kotani, M.: The first eigenvalue of homogeneous minimal hypersurfaces in a unit sphere S n+1(1). Tôhoku Math. J. 37, 523–532 (1985)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Milnor, J.: On manifolds homeomorphic to the 7-sphere. Ann. Math. (2) 64, 399–405 (1956)Google Scholar
  28. 28.
    Miyaoka, R.: Transnormal functions on a Riemannian manifold. Differ. Geom. Appl. 31, 130–139 (2013)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Miyaoka, R.: Isoparametric hypersurfaces with (g, m) = (6, 2). Ann. Math. 177, 53–110 (2013)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Montiel, S., Ros, A.: Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math. 83, 153–166 (1986)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Muto, H.: The first eigenvalue of the Laplacian of an isoparametric minimal hypersurface in a unit sphere. Math. Z.197, 531–549 (1988)Google Scholar
  32. 32.
    Muto, H., Ohnita, Y., Urakawa, H.: Homogeneous minimal hypersurfaces in the unit sphere and the first eigenvalue of the Laplacian. Tôhoku Math. J. 36, 253–267 (1984)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Münzner, H.F.: Isoparametric hyperflächen in sphären, I and II. Math. Ann. 251, 57–71 (1980) and 256, 215–232 (1981)Google Scholar
  34. 34.
    Qian, C., Tang, Z.Z., Yan, W.J.: New examples of Willmore submanifolds in the unit sphere via isoparametric functions, II. Ann. Glob. Anal. Geom. 43, 47–62 (2013)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Qian, C., Tang, Z.Z.: Isoparametric functions on exotic spheres. arXiv:1303.6028Google Scholar
  36. 36.
    Solomon, B.: Quartic isoparametric hypersurfaces and quadratic forms. Math. Ann. 293, 387–398 (1992)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Straume, E.: Compact connected Lie transformation groups on spheres with low cohomogeneity, I. Mem. Am. Math. Soc. 119(569), 1–93 (1996)MathSciNetGoogle Scholar
  38. 38.
    Tang, Z.Z., Yan, W.J.: New examples of Willmore submanifolds in the unit sphere via isoparametric functions. Ann. Glob. Anal. Geom. 42, 403–410 (2012)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Tang, Z.Z., Yan, W.J.: Isoparametric foliation and Yau conjecture on the first eigenvalue. J. Differ. Geom. 94, 521–540 (2013)MATHMathSciNetGoogle Scholar
  40. 40.
    Tang, Z.Z., Zhang, W.P.: η-invariant and a problem of Berard-Bergery on existence of closed geodesics. Adv. Math. 254, 41–48 (2014)Google Scholar
  41. 41.
    Tang, Z.Z., Xie, Y.Q., Yan, W.J.: Schoen-Yau-Gromov-Lawson theory and isoparametric foliations. Commun. Anal. Geom. 20, 989–1018 (2012)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Tang, Z.Z., Xie, Y.Q., Yan, W.J.: Isoparametric foliation and Yau conjecture on the first eigenvalue, II. J. Funct. Anal. 266, 6174–6199 (2014)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Thorbergsson, G.: A survey on isoparametric hypersurfaces and their generalizations. In: Handbook of Differential Geometry, vol. I, pp. 963–995. North-Holland, Amsterdam (2000)Google Scholar
  44. 44.
    Wang, Q.M.: Isoparametric functions on Riemannian manifolds. I. Math. Ann. 277, 639–646 (1987)CrossRefMATHGoogle Scholar
  45. 45.
    Yau, S.T.: Problem section. In: Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102. Princeton University Press, Princeton (1982)Google Scholar

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© Springer Japan 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical Sciences, Laboratory of Mathematics and Complex SystemsBeijing Normal UniversityBeijingChina

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