229 problems

For $n\ge 3$, let $g$ be a smooth Riemannian metric on the torus $\mathbb{T}^n$. Suppose that every primitive class $\alpha\in H_{n-1}(\mathbb{T}^n;\mathbb{Z})$ is represented by a…

Let $m\ge 2$, let $g$ be a Riemannian metric on $\mathbb{R}\mathbb{P}^{m}$, and let $\operatorname{sys}(g)$ denote the length of the shortest loop representing the nontrivial eleme…

For every integer $n\geq 2$, there exists a constant $C_n>0$ such that, for every closed oriented $n$-manifold $M$, every Riemannian metric $g$ on $M$, and every $\lambda\geq 0$, t…

For every $n,m\ge 1$, the set of possible constant values of $S=\lvert h\rvert^2$ for closed, minimally immersed $n$-dimensional submanifolds $M^n\hookrightarrow \mathbb{S}^{n+m}$…

For every complete Riemannian manifold $(M,g)$ with nonnegative Ricci curvature, $\operatorname{Ric}_g\ge 0$, the fundamental group $\pi_1(M)$ is finitely generated.

Conjecture 1.3 (Brin--Karcher, 1984). Let $M$ be a compact connected manifold with a $C^3$ Riemannian metric of negative variable sectional curvature $K$, satisfying…

Let $f^\beta(x)$ be the unique solution of equation (2) with $f^\beta(0)=1$. Asymptotic growth conjecture. As $x\to+\infty$, the function $f^\beta$ should be asymptotic to … for a…

Let $(M^n,g)$ be a Riemannian manifold with $n\geq 2$, and write $g=(g_{ij})$ in local coordinates $(x^1,\ldots,x^n)$. Let $t$ denote time, and let $\xi^1,\ldots,\xi^n$ be arbitrar…

Octonionic Calabi–Yau conjecture. There exist a $C^\infty$-smooth function $u\colon M\longrightarrow\mathbb{R}$ and a constant $A>0$ such that

Polynomiality conjecture. Each coefficient $a^{i_1i_3\cdots i_{2l-1}}_{i_2i_4\cdots i_{2l}}$ is a polynomial in $\theta$ of degree $l$. Conjecturally, the constant in the displayed…

Let $p\ge 2$ and let $\phi:U\to\mathbb{C}$ be a complex-valued $(p,1)$-harmonic morphism defined locally on the standard Euclidean space $\mathbb{R}^{2p-1}$. The nonexistence conje…

Extension conjecture. There are an open neighborhood $U\subset\Bbb C^n$ of $0$ and a Kronecker web $\cal W'$ on $U$ whose $k$-jet at $0$ agrees with $\cal W$.

Analytic approximation conjecture. There is an open subset $U'\subset U$ containing $0$ and a real-analytic Kronecker web $\cal W'$ on $U'$ whose $k$-jet at $0$ agrees with the $k$…

Let $M$ be a closed smooth $4$-manifold. Uniqueness conjecture. $M$ admits Einstein metrics for at most one sign of the scalar curvature. The conjecture proposes that, for a fixed…

Quantum-layer discrete-spectrum conjecture. If $\varSigma$ is not the plane and $K$ is integrable on $\varSigma$, then the Dirichlet Laplacian on $\Omega$ has non-empty discrete sp…

Turbiner's second conjecture. If the symbol of $H$ engenders a Euclidean geometry, then the spectral equation $H\psi=E\psi$ can be solved by separation of variables. The paper stat…

A quasi-exactly-solvable or exactly-solvable problem in $\mathbb{R}^2$ is one containing the Laplace–Beltrami operator with a flat-space metric tensor; its variables are non-separa…

Genus-zero conjecture. $M$ is a Riemann minimal example.

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

Existence conjecture. There exists a complex-valued harmonic morphism

Let $(M^m,g)$ be an irreducible Riemannian symmetric space of dimension $m\ge 2$. For each point $p\in M$, a complex valued harmonic morphism is a harmonic morphism … defined on an…

Burghelea–Haller conjecture. The Burghelea–Haller quadratic form satisfies

Let $\Sigma$ be an embedded asymptotically flat surface in $R^3$ that is not totally geodesic, and suppose its Gauss curvature is integrable. Consider the quantum layer built from…

Let $X$ be a closed oriented $4$-manifold with $b_2^+(X)=1$. The period map sends a Riemannian metric to the self-dual line $H^+$ in the cone $\mathcal{P}$ of elements of positive…

Let $M^{2n-1}\be2 {\bf R}^{2n}$ be a non-centrally symmetric hypersurface. A skew brane is an $(2n-1)$-dimensional submanifold of ${\bf R}^{2n+1}={\bf R}^{2n}\be2 {\bf R}$; the ver…