93 problems

A proper biharmonic submanifold is a biharmonic submanifold that is not minimal. A submanifold has constant mean curvature when the length of its mean-curvature vector is constant.…

Let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, let $\Pi_{2k+1}:\mathbb{S}^{2k+1}\to\gamma_{2k+1}(\mathbb{S}^1)$ be the neares…

Let $\varphi:M^m\hookrightarrow\mathbb{S}^{m+1}$ be a biharmonic submanifold, and let $\mathbf H$ denote its mean curvature vector field. The mean curvature has constant length whe…

A prime closed geodesic on a Finsler manifold is a closed geodesic that is not an iteration of another closed geodesic. For a Finsler metric on the sphere $S^n$, let $[x]$ denote t…

Let $M^n$ be a closed embedded minimal hypersurface of the sphere $\mathbb{S}^{n+1}$, and let $\lambda_1(M)$ denote the first positive eigenvalue of its Laplacian. Yau's conjecture…

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Write $\operatorname{conn}(K)$ for the connectivi…

Let $S^2_r$ be the 2-dimensional sphere of radius $r>1/2$, and let a three-coloring assign one of three colors to every point of $S^2_r$. Simmons's conjecture. Every such coloring…

Let $\mathcal{M}^n\subset \mathbb{S}^{n+1}$ be a compact, embedded minimal hypersurface, and let $\lambda_1(\mathcal{M}^n)$ denote the first nonzero eigenvalue of its Laplace opera…

Let $\mathbb{S}^2$ be the two-dimensional sphere, and let $\Lambda_k(\mathbb{S}^2)$ denote the supremum of the normalized $k$-th non-zero Laplace eigenvalue over metrics on…

Spherical Erdős distance conjecture. On $S^2$, the Erdős distance conjecture should hold without the logarithmic factor; that is, $A$ should determine at least a constant times $N$…

Proposed conjecture. For $n\equiv 6\pmod 8$ with $n\geq 14$,

An odd-dimensional sphere is the standard sphere $S^{2n+1}$, and a parallelizable manifold is a manifold with trivial tangent bundle. A Sasakian-Einstein metric is a Sasakian metri…

Let $\mathbb S^{l}(\frac{1}{2})\times\mathbb S^{l}(\frac{1}{2})$ be the generalised Clifford torus, viewed as a submanifold of $\mathbb S^{2l+2}$, and consider the associated bihar…

Orbital diameter conjecture. The orbital diameter satisfies

Let $S^{n-1}$ be the unit sphere, and let a positive definite function on $S^{n-1}$ be understood in the usual sense for functions on the sphere. Damase's continuity approximation…

Let $S^k$ be a $k$-sphere, and let $X(S^k)$ denote its chromatic number. Knill's sphere-coloring conjecture. Every $k$-sphere satisfies … This conjecture concerns the possible chro…

Let $T_1$ be a triangulation of the $n$-sphere and let $S_{n+2}^n$ be the boundary triangulation of the $(n+1)$-simplex. For a coloring $L:V(T_1)\to V(S_{n+2}^n)$, let…

Let $\mathbb{F}_q$ be a finite field, let $d$ be odd, and for $j\in\mathbb{F}_q$ define the sphere … where $\|x\|=x_1^2+\cdots+x_d^2$. Let $E\subseteq S_j$ with $j\neq 0$. HIKR sph…

Let $S^m$ be the standard sphere, let $1\le p<\frac{2m}{m-1}$, and let $\mathcal{A}=\{A_\lambda\}_{\lambda\ge1}$ be a family of subsets of $S^m$. An $L^p$-Logvinenko-Sereda family…

Let $m$ be the mass of a material point moving on the sphere $S^{n-1}\subset\mathbb{R}^n$, let the point be placed in a constant homogeneous magnetic field described by the paramet…

Let $S^{n-1}\subset\mathbb{R}^n$ be the sphere, and consider the magnetic system obtained by restricting the motion in a constant homogeneous magnetic field to $S^{n-1}$. Integrabi…

Let $\varphi:M^m\hookrightarrow\mathbb{S}^{m+1}$ be a biharmonic hypersurface, and let $\mathbf H$ denote its mean curvature vector. The hypersurface has constant mean curvature (C…

The restriction conjecture for the sphere. This estimate should hold when

For each dimension $n\ge 1$ and scale $r>0$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r$. Finite CW-complex conjecture…

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Countable homotopy-type changes conjecture. The h…