34 problems

For $g\geq 2$, let $\M_g$ be the moduli orbifold of genus-$g$ Riemann surfaces. Nonpositive-curvature conjecture. The orbifold $\M_g$ admits no complete, finite-volume Riemannian m…

Let $M$ be a closed, nonpositively curved, locally symmetric manifold with no local $\mathbb{R}$ factors. The Gromov norm of $M$, namely the simplicial volume, is positive. Gromov'…

Let $M$ be a closed $n$-manifold with nonpositive sectional curvature, negative Ricci curvature, and no local $\mathbb{R}$ factors. Let $N$ be any closed Riemannian manifold and le…

Let $M$ be a closed manifold which admits a locally symmetric Riemannian metric $g_{loc}$ with nonpositive sectional curvature. Assume that $(M,g_{loc})$ has no local factors isome…

Let $M^d$ be a compact, closed, $d$-dimensional manifold with a piecewise Euclidean structure, and suppose that it is non-positively curved in the sense of Gromov. Let $\chi(M^d)$…

Euler characteristic conjecture.

Let $(X,g)$ be a closed $2n$-dimensional Riemannian manifold with sectional curvature $\mathrm{sec}_{g}$. Write $\chi(X)$ for its Euler characteristic. Hopf conjecture. … and … Thi…

Let $M$ be a biharmonic submanifold of a non-positively curved Riemannian manifold. Generalized Chen's conjecture. The submanifold $M$ is minimal. Ou and Tang constructed counterex…

Let $M$ be a closed nonpositively curved $4$-manifold, and let $\|M\|$ denote its simplicial volume and $\chi(M)$ its Euler characteristic. Connell–Ruan–Wang's equivalence conjectu…

Let $M$ be a closed, connected, oriented $n$-dimensional manifold. Its simplicial volume is … A Riemannian metric on $M$ has negative definite Ricci curvature when its Ricci curvat…

Let $M$ be a manifold. Suppose $M$ admits a Riemannian metric with nonpositive sectional curvature everywhere and negative definite Ricci curvature at some point. Connell–Wang's co…

Let ${{\normalfont\textsc{Poly}}}_d^{mt}$ be the space of all monic degree-$d$ complex polynomials up to translation with its stratified Euclidean metric, and let $K_d$ be the dual…

Let ${{\normalfont\textsc{Poly}}}_d^{mt}$ be the space of all monic degree-$d$ complex polynomials up to translation, equipped with its stratified Euclidean metric. CAT(0) conjectu…

Let $A_S$ be an Artin group and let $S'\subset S$ be an irreducible almost spherical subset. Equip the relative Artin complex $\Delta_{S,S'}$ with the metric described in the sourc…

Flat-geodesic finiteness conjecture. All flat geodesics are closed, and there are only finitely many homotopy classes of such geodesics. In particular,

Let $M$ be a closed nonpositively curved $4$-manifold, with simplicial volume $||M||$ and Euler characteristic $\chi(M)$. Zero simplicial volume characterization conjecture. Then ……

Let $M$ be a closed, connected, oriented topological $n$-manifold. Suppose that $M$ admits a Riemannian metric whose sectional curvature is nonpositive everywhere and whose Ricci c…

Let $X$ be a compact, two-dimensional, non-positively curved space. Finite complex model conjecture. The space $X$ is homotopy equivalent to a finite, two-dimensional, non-positive…

Let $(M,g)$ be a smooth, connected and closed surface of genus $d524\geq 2$ with nonpositive curvature. Let $SM$ be its unit tangent bundle, let $\Lambda$ denote the set of flat ge…

Visibility-end conjecture. The fundamental group of each end of $M$ is almost nilpotent.

Let $M$ be a nonpositively curved compact Kähler manifold. Zheng's conjecture. If $M$ is of general type, then it must be Kobayashi hyperbolic, meaning that every holomorphic map ……

Strengthened Gromov conjecture. Then $M$ has positive simplicial volume:

Let $M$ be a tame, complete, finite-volume $n$-manifold of bounded nonpositive curvature. The thick-thin conjecture. There is a compact subset $C\twoheadrightarrow M$ that cannot b…

Let $M$ be a tame, complete, finite-volume $n$-manifold of bounded nonpositive curvature. A manifold has geometric rank one when it satisfies the rank-one condition used in the sou…

Let $M$ be a Riemannian manifold with non-negative Ricci curvature, and let $X$ be an NPC metric space. A harmonic map is a map $u:M\rightarrow X$ satisfying the harmonic-map condi…