9 problems

Proper Dupin hypersurface conjecture. Every compact embedded proper Dupin hypersurface in $S^5$ with four distinct principal curvatures is diffeomorphic to

Let $M^n$ be a closed, minimally immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. Stronger version of Chern's conjecture. Then $M^n$ is i…

Bryant's conjecture. $M$ is isoparametric.

Second pinching conjecture. If $S>n$, then $S\geq 2n$.

Let $M^{n}$ be a closed immersed minimal hypersurface of the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. A hypersurface is isoparametric if it has constant princ…

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$, with $g$ distinct principal curvatures. The hypersurface has constant Lie curvatures when its Lie curvatures are…

Let $M^n$ be a closed immersed minimal hypersurface of the sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. Refined Chern conjecture. Then $M^n$ is isoparametric. The text…

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$ with either four or six principal curvatures. Assume that its Lie curvatures are constant; these are the Lie-sphe…

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$ or ${\bf R}^n$. Cecil–Ryan conjecture. $M$ is Lie equivalent to an isoparametric hypersurface in $S^n$. The conje…