960 problems

Let $(X,g)$ be a closed, $2n$-dimensional Riemannian manifold with negative sectional curvature $\mathrm{sec}_{g}$. Let $h^k_{(2)}(X)$ denote the $k$-th $L^2$-Betti number of $X$.…

Lichnerowicz's conjecture. There are no other simply connected harmonic manifolds.

Let $M$ be a closed smooth manifold, and let $g$ be an $L^\infty$-metric on $M$, meaning that for some smooth reference Riemannian metric $g_0$ and constant $\Lambda>0$,…

A positive quaternion-Kähler manifold is a complete quaternion-Kähler manifold $(M^{4n},g)$ with positive scalar curvature; the known examples are the Wolf spaces, which are compac…

A biharmonic submanifold is a submanifold whose inclusion map is biharmonic; a minimal submanifold has vanishing mean curvature. Let a submanifold lie in a Riemannian manifold whos…

A simple Riemannian manifold is a compact Riemannian manifold with strictly convex boundary and no conjugate points, such that every pair of points is joined by a unique minimizing…

A homogeneous expanding Ricci soliton is a homogeneous Riemannian manifold whose Ricci flow evolves by scaling and diffeomorphisms with an expanding scale factor. Generalized Aleks…

Let $g$) be a smooth Riemannian metric on the closed hemisphere $\mathbb{S}^3_+$ satisfying the following conditions: its scalar curvature obeys ${\mathrm R}_g\geq 6$; its induced…

Let $M$ be an open manifold with nonnegative Ricci curvature, and let $\widetilde M$ be its universal cover. The manifold $\widetilde M$ has Euclidean volume growth if there is a c…

Let $(M,g,f)$ be a compact gradient Ricci soliton with positive curvature operator. A gradient Ricci soliton is a Riemannian manifold whose Ricci tensor satisfies an equation of th…

Let $(M^n,g,f)$ be a CPE metric, meaning that $(M^n,g)$ is a closed, oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature and $f:M^n\to\mathbb{R}$ is…

A Cartan-Hadamard manifold $M^n$ is a complete simply connected Riemannian manifold of nonpositive sectional curvature. For a bounded region $a9\subset M^n$ of finite perimeter, wr…

Geroch's conjecture. Any Riemannian metric on a torus with non-negative scalar curvature must be flat.

Nishikawa's conjecture. If $M$ has positive curvature operator of the second kind, then $M$ is diffeomorphic to a spherical space form; if $M$ has nonnegative curvature operator of…

Let $M$ be an irreducible symmetric space of compact type and rank greater than one. A polar action on $M$ is an isometric group action admitting a section, namely a complete immer…

Let $(\Omega^{n+1},g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$. Assume that the Ricci curvature of $\Omega$ satisfies…

Let $\mathcal{M}$ be a homogeneous quaternionic Kähler manifold with negative Ricci curvature. The manifold is called normal if it admits a transitive solvable group of isometries…

Schläfli's conjecture. The system is locally solvable provided that the target Euclidean space has dimension $d\geq n_{\ast}$. This concerns the existence of local isometric immers…

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$, and let $\omega_k(M^{n+1},g)$ be its min–max widths. For a generic metric $g$, write a min–max realization…

Let $M$ be a compact Riemannian manifold, and let $K$ be a curvature formed locally from the Riemannian curvature tensor, its covariant derivatives, and the metric. A conformal inv…

A Riemannian manifold $(M,\langle\,\ \rangle)$ is globally Osserman if the eigenvalues of its Jacobi operator $R_u$ are independent of both the unit tangent vector $u\in M_x$ and t…

Gromov's quantitative scalar-curvature conjecture. There exist constants $C(g_0)$ and $\varepsilon_0(g_0)$ such that, whenever $\varepsilon\leq \varepsilon_0$, one has

Homogeneity Conjecture. The manifold $(M',ds'^2)$ is homogeneous if and only if every $\gamma\in\Gamma$ is an isometry of constant displacement on $(M,ds^2)$.

Let $X$ be a manifold, and let a complete positive-scalar-curvature (PSC) metric mean a complete Riemannian metric whose scalar curvature is everywhere positive. Rosenberg–Stolz co…

Let $(M^n,g,f)$ be an $n$-dimensional compact Vacuum Static Space, meaning that $(M^n,g)$ is an $n$-dimensional Riemannian manifold and $f$ is a non-constant smooth function satisf…