22 problems

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 $M$ be a closed embedded Dupin hypersurface, meaning that each principal curvature has constant multiplicity and is constant along its curvature direction. A Lie image is the i…

Let $M$ be a hypersurface in the Euclidean sphere. Its mean curvature and scalar curvature are the corresponding curvature functions of $M$, and it is isoparametric when its princi…

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…

Strong Chern's conjecture. $M^n$ is isoparametric.

Let $M$ be a closed minimal hypersurface in the unit sphere $\mathbb S^{n+1}$ with constant scalar curvature. A minimal hypersurface is isoparametric when its principal curvatures…

Let $M$ be an irreducible connected proper Dupin hypersurface in $S^n$ with four principal curvatures. Suppose their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and let $\Psi$…

An isoparametric hypersurface in a sphere is a hypersurface with constant principal curvatures; let $g$ denote the number of distinct principal curvatures. Dorfmeister–Neher conjec…

For an isoparametric hypersurface with $g=6$ principal curvatures, all principal curvatures have the same multiplicity $m$. In the case $m=2$, there is one known homogeneous family…

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

An isoparametric hypersurface with $g=4$ principal curvatures has multiplicities equal to those of a known Ferus–Karcher–Münzner example or one of the two homogeneous exceptions wi…

Let $x\colon M^3\longrightarrow\Bbb{S}^4(1)$ be a hypersurface immersion, with normalized mean curvature $H$, second elementary symmetric curvature function $H_2$, and Gauß–Kroneck…

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 $M\subset S^n$ be a compact proper Dupin hypersurface, meaning that it has constant multiplicities of its principal curvatures along each curvature surface. Two hypersurfaces a…

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 $M^m$ be a hypersurface in the unit sphere ${\mathbb S}^{m+1}$. Here $r$-harmonic and $ES$-$r$-harmonic refer to the notions used in the source, and proper means $r$-harmonic b…

Let $(M^n,g)$ be a closed embedded minimal hypersurface in the unit sphere $S^{n+1}(1)$, and let $\lambda_1(M)$ denote the first eigenvalue of its Laplace–Beltrami operator. Yau's…

Let $M^{n-1}\subset S^n$ be a compact, connected proper Dupin hypersurface. Let $g$ be the number of distinct principal curvatures, and call its Lie curvatures constant Lie curvatu…

Let $M^{n-1}subset S^n$ be a compact, connected proper Dupin hypersurface, and let $g$ denote its number of distinct principal curvatures. Dupin classification conjecture. Every su…