237 problems

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 $\Lambda>0$,…

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…

Volume rigidity conjecture. The metric $g$ is isometric to the hyperbolic metric $g_H$.

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…

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

A closed aspherical manifold is a closed manifold whose universal cover is contractible. Gromov–Lawson conjecture. A closed aspherical manifold cannot support a Riemannian metric o…

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

Sormani's conjecture. If $g$ satisfies the min-A condition, then $(T^n,g)$ should be close to a flat torus in the intrinsic flat topology.

Let $g$ be a smooth complete metric on $\mathbb R^n$, where $n\geq 3$, and let $g_{\mathrm{euc}}$ denote the Euclidean metric. Assume that … and … as $x\to\infty$. Gromov's Euclide…

Rosenberg's $S^1$-stability conjecture.

Gromov's band-width conjecture. The metric $g$ satisfies

Boundary curvature-integral conjecture. One has

Gromov–Sormani MinA scalar compactness conjecture. A subsequence converges in the volume-preserving intrinsic flat sense to a three-dimensional rectifiable limit space $M_\infty$.…

For a unit vector $cvec{a}=(a^1,a^2,a^3)$ and $s oindenttext{in}cmathbb{R}$, let … be the totally umbilic planes in the upper-half-space model $(cmathbb{R}^3_+,b)$ of hyperbolic…

Let $(M^3,g)$ be a Riemannian three-manifold with scalar curvature $R_M\geq 0$, and let $\Sigma\subset M$ be a two-sided two-torus. A surface is locally area-minimizing if it minim…

Let $M$ be a manifold of dimension $m$, and let $T^m$ denote the $m$-dimensional torus. Generalized Geroch conjecture. There is no complete metric of positive scalar curvature on ……

Yau's conjecture. The scalar curvature should satisfy

Let $W$ be a closed $m$-dimensional manifold such that $W$ minus a point does not admit a complete metric of positive scalar curvature. Let $M$ be obtained from $W$ by removing a s…

Let $\mathcal{H}_\xi$ be the space of holomorphic maps associated with parameter $\xi$, and let $\mathcal{M}_\xi$ be the corresponding vortex moduli space. Assume that…

Gromov's conjecture. There exists a dimensional constant $C(n)$ such that, for every $p\in M$,

Let $(M^n,g)$ be a complete Riemannian manifold with scalar curvature … Here $\operatorname{FillRad}(M)$ denotes the filling radius, and $c_n$ is a universal constant depending onl…

Let $P$ be a convex $n$-dimensional polytope in $\mathbb R^n$, equipped with a Riemannian metric $g$. Assume that each face is mean convex with respect to $g$, the dihedral angles…

Let $\Sigma$ be the boundary of a compact Riemannian manifold $\Omega$. Suppose that the scalar curvature $R$ of $\Omega$ satisfies … for a constant $\sigma$. Gromov's conjecture.…

Marques–Neves' width stability conjecture. Then $(\mathbb S^3,g_k)$ converges in the Sormani–Wenger intrinsic flat sense to the round unit sphere $(\mathbb S^3,g_{rd})$. The conjec…

Let $(M,g)$ be a Riemannian manifold of dimension $3\leq n\leq 7$ that is asymptotically flat of rate $\tau=n-2$, with non-negative scalar curvature. Suppose that $M$ contains a no…