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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 18, Issue 2

Issues

On CMC hypersurfaces in 𝕊n+1 with constant Gauß–Kronecker curvature

S. C. de Almeida
  • CAEN, Universidade Federal do Ceará, Av. da Universidade, 2700 - 2 andar - Benfica, 60020-181 Fortaleza-CE, Brazil
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/ F. G. B. Brito
  • Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09.210-170 Santo AndrĂ©, Brazil
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/ M. Scherfner
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  • Institute of Mathematics and its Didactics, Bochum University of Applied Sciences, Lennershofstr. 140, 44801 Bochum, Germany
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/ S. Weiss
Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/advgeom-2017-0054

Abstract

After nearly 50 years of research the Chern conjecture for isoparametric hypersurfaces in spheres is still an unsolved and important problem. Here we give a partial result for CMC hypersurfaces with constant Gauß–Kronecker curvature, mainly using a result given in [3] by Otsuki.

Keywords: CMC hypersurface; constant Gauß–Kronecker curvature; Chern conjecture; isoparametric hypersurface

MSC 2010: 53C42; 53C40

References

  • [1]

    R. Miyaoka, Complete hypersurfaces in the space form with three principal curvatures. Math. Z. 179 (1982), 345–354. MR649037 Zbl 0465.53037CrossrefGoogle Scholar

  • [2]

    R. Miyaoka, Correction of complete hypersurfaces in the space form with three principal curvatures. Bol. Soc. Brasil. Mat. 18 (1987), 83–94. MR1018447 Zbl 0825.53002CrossrefGoogle Scholar

  • [3]

    T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature. Amer. J. Math. 92 (1970), 145–173. MR0264565 Zbl 0196.25102CrossrefGoogle Scholar

  • [4]

    M. Scherfner, S. Weiss, S.-T. Yau, A review of the Chern conjecture for isoparametric hypersurfaces in spheres. In: Advances in geometric analysis, volume 21 of Adv. Lect. Math. (ALM), 175–187, Int. Press, Somerville, MA 2012. MR3077256 Zbl 1320.53023Google Scholar

About the article


Received: 2016-03-23

Published Online: 2018-01-24

Published in Print: 2018-04-25


Citation Information: Advances in Geometry, Volume 18, Issue 2, Pages 187–192, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0054.

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