57 problems

Gromov's conjecture. $\mathbb{Z}_2$-systolic freedom does not exist.

Loewner's bound. Every aspherical surface satisfies

Systolic-category cup-length conjecture. One should have

The systolic-category lower-bound conjecture. Such a manifold should satisfy

Relative Pu conjecture. There is a point $p\in X$ such that

Pu–Loewner equality rigidity conjecture. Both the topology and the metric must be extremal: (a) $X$ fibers smoothly over $\mathbb T^2$ with fiber $\mathbb R P^2$; (b)…

Pu–Loewner inequality conjecture. Every such metric satisfies

Universal stable 2-systolic inequality conjecture. There is a numerical constant $C$, independent of $X$, such that for every metric $\mathbf{g}$ on $X$,

Pu's relative 1-systole conjecture. For any such $S$ and $\hat\phi$,

Fix a symmetric space $S$ of non-compact type without Euclidean de Rham factors, and let an arithmetic $S$-manifold be a quotient $M=\Gamma\backslash S$ by a torsion-free arithmeti…

Let $N\geq 2$, let $1\leq k\leq 2N-2$, set $d=k+1$, and let $\mu>0$. For the Berger metric $g_\mu$ on $\mathbb {RP}^{2N-1}$, define the mod-two homological systole by … For a real…

Let $(X,L)$ be a principally polarized complex abelian variety of dimension $3$, and let $\lambda_1(X,L)$ denote the length of its shortest closed geodesic, equivalently twice the…

Let $\Sigma$ be a closed oriented smooth manifold of dimension $2n-1$, with $n\geq 2$, and let $C_0\in H^2(\Sigma,\mathbb{R})$ be represented by a Zoll odd-symplectic form…

Let $W$ be a closed, connected smooth manifold of dimension $n$, and for a smooth Riemannian metric $g$ on $W$ let … where $\operatorname{sys}(g)$ is the length of the shortest non…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type product inequality asserts that there is a constant $C(n)>0$, depending only on the dimensi…

Let $X$ be a translation surface. Write $\mathop{\mathrm{KVol}}(X)$ for its KVol and $\mathop{\mathrm{SysVol}}(X)$ for its systolic volume, defined as the volume divided by the squ…

Let $\mathcal{H}$ be a smooth Riemannian metric on the circle $\mathbb{S}^{1}$, and let $FillVol(\mathbb{S}^{1},\mathcal{H})$ denote the infimum of the areas of compact orientable…

Homotopical $2$-systole conjecture. Under these hypotheses,

Contractible systolic-ratio conjecture. Zoll metrics on $S^2$ are global maximizers of the contractible systolic ratio among metrics on $S^2$.

Croke's conjecture. The optimal upper bound for the systolic ratio of Riemannian $2$-spheres is

Croke's conjecture. The extremal metric is the Calabi–Croke doubled triangle, obtained by gluing two flat equilateral triangles with side lengths $1$ along their edges.

Let $\mathcal R(S^2)$ denote the space of Riemannian metrics on the two-sphere, and let $\rho(g)$ be its weak systolic ratio. The Calabi–Croke metric $g_{\Delta}$ is obtained by gl…

Let $S^2$ be the sphere equipped with a Riemannian metric of fixed area, and let ${\rm scg}(S^2)$ denote the length of its shortest closed geodesic. The Calabi–Croke sphere conject…

Let $S^2$ be the 2-sphere with an arbitrary metric, let $\area(S^2)$ denote its area, and let $\operatorname{LCG}(S^2)$ denote the least length of a nontrivial closed geodesic in…

Let $M$ be a closed, irreducible $3$-manifold with infinite fundamental group. The surface subgroup conjecture. The manifold $M$ contains an immersed $\pi_1$-injective closed surfa…