111 problems

Gromov–Lawson–Rosenberg conjecture. The manifold $M$ admits a psc-metric if and only if its Rosenberg index vanishes.

Let $M$ be a closed positive scalar curvature $n$-manifold, and let $\widetilde M$ be its universal covering equipped with the metric lifted from $M$. The macroscopic dimension…

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 $M$ be a spin manifold with higher $alpha$-invariant, and let $\mathrm{Bott}$ denote the Bott manifold. Stable Gromov-Lawson-Rosenberg conjecture. If the higher $alpha$-invaria…

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

Unstable Gromov–Lawson–Rosenberg conjecture. The vanishing of $\alpha(M)$ decides whether $M$ admits a metric of positive scalar curvature.

Let $M$ be a smooth spin manifold, and let $\alpha(M)\in \operatorname{KO}_m(\mathbb{C}^*\pi_1(M))$ denote its Rosenberg index, where $m=\dim M$. Rosenberg's universal index obstru…

Let $M$ be a uniformly contractible Riemannian manifold with bounded geometry. Gromov's conjecture. $M$ cannot have uniformly positive scalar curvature. The conjecture is presented…

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 ……

Let $M$ be a closed positive scalar curvature $n$-manifold with fundamental group $\Gamma$, and let $B\Gamma$ be a classifying space for its universal covering. The manifold $M$ is…

Let $M$ be a complete non-compact 3-manifold with scalar curvature $R\geq 6$, and let $f:M\to\mathbb R$ be the proper Morse function whose level-set components have controlled area…

Let $N$ be a closed manifold of dimension at least $5$. The Rosenberg index is the index-theoretic obstruction to positive scalar curvature obtained from the Dirac operator with co…

Gromov's rational homology vanishing conjecture. The induced map on fundamental classes satisfies

Let $n$ be a positive integer. A closed manifold is aspherical if its universal cover is contractible. A Riemannian metric has positive scalar curvature when its scalar curvature i…

Gromov's band width conjecture. There exists a constant $C_n$ such that

Farb–Weinberger conjecture. $M$ does not admit a finite-volume Riemannian metric in the quasi-isometry class of the Teichmüller metric whose scalar curvature is uniformly bounded a…

Let $M^m$ be a closed spin manifold with fundamental group $\pi$, and let $u:M\to B\pi$ classify its universal covering. The index class is an element … A group is torsion free whe…

Conformal concordance conjecture. If $C_0$ and $C_1$ are conformally concordant, then they are isotopic in ${\cal C}^+(M)$.

Let $M$ be a spin manifold with finite fundamental group of odd order, let $\widetilde M$ be its universal cover, and let $\alpha(\widetilde M)$ and $\alpha(M,u)$ denote the corres…

Gromov's simplicial volume vanishing conjecture. If $M$ admits a metric of positive scalar curvature, then

Schoen's codimension-three conjecture. Singular sets of codimension at least three should be invisible from the point of view of positive scalar curvature: they should not evade th…

Let $M$ be an orientable $n$-manifold equipped with a complete Riemannian metric of positive scalar curvature. For a complete Riemannian metric on $M$, say that it has at most $C$-…

Let $M$ be a compact manifold with boundary, and let $DM$ denote its double, obtained by gluing two copies of $M$ along their common boundary. A Riemannian metric on $M$ has positi…

Bounded-geometry positive scalar curvature exact-sequence conjecture. There is a commutative diagram

Let $M$ be a closed aspherical manifold, meaning that its higher homotopy groups satisfy $\pi_k(M)=0$ for all $k\ge2$. A positive scalar curvature metric is a Riemannian metric who…