30 problems

Let $M$ be a closed minimal hypersurface in a unit sphere, and consider the scalar curvature of $M$. Chern's conjecture. The possible constant values of the scalar curvature of clo…

Let $T\subset S^3$ be an embedded minimal torus, and let $T_{\mathrm{Cl}}$ denote the Clifford torus in $S^3$. Lawson's conjecture. There exists an isometry $F$ of $S^3$ such that…

Let $M$ be a biharmonic submanifold of a sphere. BMO conjecture. The submanifold $M$ has constant mean curvature. The source presents this as one of the well-known open conjectures…

Yau's conjecture. The volume of one of the minimal Clifford tori $M_{k,n-k}$ gives the lowest volume among all non-totally geodesic closed minimal hypersurfaces of…

Chang--Yang conjecture. For every $\alpha\geq\frac12$ and every $u$ satisfying

Let $S^{d-1}$ be a unit sphere in $\mathbb R^d$. Suppose we are given $d$ points $x_1, \ldots, x_d\in S^{d-1}$ and a continuous function $f: S^{d-1}\to \mathbb R$. Knaster's conjec…

A hypertorus is an immersion of ${\mathbf{S}^{1}}\times {\mathbf{S}^{1}}\times {\mathbf{S}^{1}}$ in ${\mathbf{S}^{4}}$. Consider minimal embedded hypertori in ${\mathbf{S}^{4}}$. C…

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…

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 $\ell$ and $d$ be positive integers. For an integer $n$, let $X=\{x_1,\ldots,x_\ell\}$ be any set of $\ell$ points on the unit sphere $\mathbb{S}^{n-1}$, and let…

Classification conjecture. The only proper biharmonic hypersurfaces in $\mathbb{S}^{m+1}$ are the open parts of hyperspheres $\mathbb{S}^m(1/\sqrt{2})$ or of the standard products…

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…

Let $M^n$ be a closed immersed minimal submanifold of the unit sphere $\mathbb{S}^{n+q}$. Let $S_\alpha$ be the shape operators with respect to an orthonormal basis…

Let $\mathbb{S}^{2n+1}$ be the unit sphere with its standard Riemannian metric, and let a harmonic unit vector field mean a unit vector field that is a critical point of the energy…

Chern's conjecture. The value of $|B|^2$ must lie in a discrete subset of $\mathbb{R}$.

Let $J_{\alpha}$ be the functional on $H^2(\mathbb{S}^6)$ defined by … where $\mathrm{d}w$ is normalized Lebesgue measure and $P_6=-\Delta(-\Delta+4)(-\Delta+6)$. Let … For every…

Let $M^n$ be a closed immersed or embedded, non-totally geodesic, minimal hypersurface in $\mathbb{S}^{n+1}$. For $a\in\mathbb{S}^{n+1}$, let $\psi_a$ denote the corresponding heig…

Let $M^n$ be a closed embedded, non-totally geodesic, minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$. Let $S$ be the squared length of its second fundamental form and l…

Let $M^3$ be a minimal hypersurface in $\mathbb{S}^4$ with constant norm of the second fundamental form, without assuming that $M$ is compact. Bryant's conjecture. The hypersurface…

Let $S^2=\{u\in\mathbb{R}^3:|u|=1\}$ be the 2-dimensional sphere, and for $n\geq 2$ let $\mu_n(S^2)$ denote the maximal number of $3$-term arithmetic progressions in an $n$-element…

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 $M^m$ 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, $f$ denotes the mean curvature…

Horizontal diameter rigidity conjecture. For any singular Riemannian foliation on a unit sphere $\mathbb{S}^{n}$, we have

Hexagonal-pattern conjecture. For large $N$ and strong point interactions, the optimal pattern should be approximately hexagonal; this is not expected to hold in the weak-coupling…