TU Berlin Fakultät II
Institut für Mathematik
     

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Isoparametric Hypersurfaces

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E. Grub (secretary), Tel.: (++49 30) 314-222 51, Fax: 314-792 82

Technische Universität Berlin
School for Mathematics and Natural Sciences
Department of Mathematics, MA 8-3
Straße des 17. Juni 136
10623 Berlin
Germany


Topic

Isoparametric hypersurfaces are of great interest, in particular since E. Cartan analyzed them for the euclidean and hyperbolic space. Since 40 years there is a famous conjecture, the Chern conjecture for isoparametric hypersurfaces in spheres:

Let M be a closed, minimally immersed hypersurface of the (n+1)-dimensional sphere with constant scalar curvature. Then M is isoparametric.

It was originally proposed in a less strong version by Chern and Chern, do Carmo and Kobayashi in 1968 and 1970 respectively. So far, no proof for the conjecture has been found, although partial results exist in particular for low dimensions and with additional conditions for the curvature functions of M.

In our work we are concerned with the conjecture and generalizations. Several articles were published and - together with colleagues from Brazil and China - we try to find the next steps towards a proof.

Guests (recent)

Francois Vigneron, Univ. Paris Est

Ryszard Deszcz, Univ. Wroclaw

Frédéric Robert, Laboratoire J. A. Dieudonné

Henri Anciaux, Univ. Tours

Graham Hall, Univ. Aberdeen


Talks

  • Will be announced

Conferences (recent)

  • 33. Süddeutsches Differentialgeometrie-Kolloquium, Vienna, Austria, 2008
  • Workshop on Differential Geometry, Kunming, China, September 2008
  • The Conference on Geometry in honour of Shing-Tung Yau on his 60th birthday, Warsawa, Poland, April 2009
  • 34. Süddeutsches Kolloquium über Differentialgeometrie, Munich, Germany, June 2009

Related Publications

  • T. Lusala, M. Scherfner, L. A. M. Sousa Jr.: Closed Willmore hypersurfaces of S^5 with constant mean and scalar curvature,
    Asian J. Math. Vol. 9 No. 1 (2005)
  • M. Scherfner: The Chern conjecture for isoparametric hypersurfaces in spheres: history and new results,
    Proceedings of the "Symposium on the differential geometry of submanifolds" (2007)
  • M. Scherfner, S. Weiss: Towards a proof of the Chern conjecture for isoparametric hypersurfaces in spheres,
    Proc. 33. South German Diff. Geom. Colloq. (2008)
  • S. Weiss: Über eine Beweismethode zum Nachweis von Isoparametrie,
    Proc. 34. South German Diff. Geom. Colloq. (to appear)
  • M. Scherfner, S. Weiss: How to Prove that Closed Hypersurfaces in Space Forms of Constant Curvature are Isoparametric?
    accepted, Proc. XVI Geometrical Seminar (2011)
  • M. Scherfner, S. Weiss, S.-T. Yau: A review of the Chern conjecture for isoparametric hypersurfaces in spheres,
    submitted (2011)
  • M. Scherfner, L. Vrancken, S. Weiss: On closed minimal hypersurfaces with constant scalar curvature in S7,
    submitted (2011)
  • M. Scherfner, S. Weiss: On CMC Hypersurfaces in Spheres with Constant Gau-Kronecker Curvature and three Distinct Principal Curvatures,
    submitted (2011)
  • M. Scherfner, S. Weiss: Hypersurfaces in Spheres with Constant Mean and Scalar Curvature with three Distinct Principal Curvatures,
    submitted (2011)
  • M. Scherfner, L.A.M. Sousa Jr., S. Weiss: Closed Willmore Hypersurfaces of S5 with Constant Mean and Scalar Curvature,
    preprint (2011)

Mike Scherfner . 31.05.2011.