Compact Dupin hypersurfaces

Abstract

A hypersurface M in R is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on M , i.e., each continuous principal curvature function has constant multiplicity on M . These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in R or S. The theory of compact proper Dupin hypersurfaces in S is closely related to the theory of isoparametric hypersurfaces in S, and many important results in this field concern relations between these two classes of hypersurfaces. This problem was formulated in 1985 in a conjecture of Cecil and Ryan [17, p. 184], which states that every compact, connected proper Dupin hypersurface M ⊂ S is equivalent to an isoparametric hypersurface in S by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments. 1 Dupin hypersurfaces This paper is a survey of the main results in the theory of compact, proper Dupin hypersurfaces in Euclidean space R, which began with the study of the cyclides of Dupin in R in a book by Dupin [25] in 1822. This theory is closely related to the well-known theory of isoparametric hypersurfaces in S, i.e., hypersurfaces with constant principal curvatures in S, introduced by E. Cartan [2]–[5] and developed by many mathematicians (see [7], [23], [65], [18, pp. 85–184] for surveys).