Compact Dupin Hypersurfaces

Authors

Thomas E. Cecil, College of the Holy CrossFollow

Document Type

Article

Publication Date

4-13-2021

Department

Mathematics

Abstract

A hypersurface M in Rⁿ 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 Rⁿ or Sⁿ . The theory of compact proper Dupin hypersurfaces in Sⁿ is closely related to the theory of isoparametric hypersurfaces in Sⁿ, and many important results in this field concern relations between these two classes of hypersurfaces. In 1985, Cecil and Ryan [18, p. 184] conjectured that every compact, connected proper Dupin hypersurface M ⊂ Sⁿ is equivalent to an isoparametric hypersurface in Sⁿ by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.

Comments

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

Repository Citation

Cecil, Thomas E., "Compact Dupin Hypersurfaces" (2021). Mathematics and Computer Science Department Faculty Scholarship. 12.
https://crossworks.holycross.edu/math_fac_scholarship/12