#### Document Type

Article

#### Publication Date

1-13-2021

#### Department

Mathematics

#### Abstract

A hypersurface *M* in **R**^{n} 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**^{n} or S^{n} . The theory of compact proper Dupin hypersurfaces in S^{n} is closely related to the theory of isoparametric hypersurfaces in S^{n}, 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^{n} is equivalent to an isoparametric hypersurface in S^{n} by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.

#### Recommended Citation

Cecil, Thomas E., "Compact Dupin Hypersurfaces" (2021). *Mathematics Department Faculty Scholarship*. 12.

https://crossworks.holycross.edu/math_fac_scholarship/12

## Comments

Link to ArXiv version: http://arxiv.org/abs/2101.05316