Dupin hypersurfaces with three principal curvatures
In this very interesting paper, the authors prove that any irreducible Dupin hypersurface in \(S^n\) with three principal curvatures is equivalent up to a Lie transformation to an isoparametric hypersurface. Note that there are no global conditions assumed in this theorem. If one replaces the word irreducible by compact in the above result, one gets a theorem of \textit{R. Miyaoka} [Math. Z. 187, 433-452 (1984; Zbl 0545.53047)]. It is a very interesting open question whether the known results on compact Dupin hypersurfaces hold true under the assumption of irreducibility instead of compactness. Here I am especially thinking of the restriction on the number of principal curvatures in my paper [\textit{G. Thorbergsson}, Bull. Lond. Math. Soc. 15, 493-498 (1983; Zbl 0529.53044)] and the recent result of \textit{S. Stolz} on the possible multiplicities of the principal curvatures [`Multiplicities of Dupin hypersurfaces' (Preprint MPI-1997-25) (Max-Planck-Institut für Mathematik, Bonn) (1997)].
scientific article; zbMATH DE number 1186158