
Geometry Group
Members
Projects
Lehre
Seminare
Images,
Videos, and Games
Virtual
Math Labs
Software
Contact



Isoparametric Hypersurfaces
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
Conferences (recent)

33. Süddeutsches DifferentialgeometrieKolloquium, Vienna, Austria, 2008

Workshop on Differential Geometry, Kunming, China, September 2008

The Conference on Geometry in honour of ShingTung 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 S^{7},
submitted (2011)

M. Scherfner, S. Weiss: On CMC Hypersurfaces in Spheres with Constant GauKronecker
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 S^{5} with Constant Mean and
Scalar Curvature,
preprint (2011)
