Compact Dupin hypersurfaces

A hypersurface $M$ in $\mathbb{R}^n$ 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 $\mathbb{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. In 1985, Cecil and Ryan [18, p. 184] conjectured that every compact, connected proper Dupin hypersurface $M \subset S^n$ 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.

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Published 18 October 2021