34 problems

Let $M^n$ be a closed manifold. In part (a), assume that $M^n$ is a flat Riemannian manifold. In part (b), assume that $M^n$ admits an almost flat structure. Let $W^{n+1}$ be a com…

Croke–Dairbekov–Sharafutdinov marked-length-spectrum volume conjecture. The inequality

Croke–Dairbekov–Sharafutdinov filling-volume conjecture. The filling volume of $\partial X$ with its chordal metric equals $\operatorname{vol}(X)$, and $X$ is the unique volume-min…

Let $(M,g)$ be a compact manifold of negative curvature without boundary, and let $(u_n)_{n\geq 1}$ be any orthogonal eigenbasis of the Laplacian. A semiclassical measure is a weak…

Connectedness conjecture. The space of Anosov metrics on $\Sigma$ is connected.

Let $M$ be a compact Riemannian manifold with negative sectional curvature and dimension at least $3$. Suppose that, for some $k\geq 2$, there exist infinitely many distinct immers…

Let $M$ be a compact Riemann manifold with negative curvature. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let $\Phi:\pi_1(M)\to\Gamma$ be surjective, and le…

Kanai's conjecture. If the distribution $\mathbb{H}$ is $\mathcal{C}^1$, then $(M,g)$ is locally symmetric.

Hopf-Chern conjecture.

Let $M$ be a pinched negatively curved manifold of infinite volume, of dimension $n$, and let $[hat{\omega}]in H^{n}_b(M,\mathbb{R})$ denote its bounded fundamental class. Kim–Kim'…

Let $X$ be a closed manifold, and let $h_1$ and $h_2$ be Riemannian metrics on $X$ with negative sectional curvature. For each such metric, let $\mathrm{MLS}_h$ assign to every con…

Let $(M^n,g)$ be a closed negatively curved manifold with higher hyperbolic rank, meaning that for every geodesic there is a nonvanishing Jacobi field whose span with the geodesic…

Let $(M,g)$ be a closed Riemannian manifold with negative sectional curvature, of dimension $n\geq 3$. An immersed totally geodesic submanifold $N$ in $M$ is maximal if it is not c…

Let $(M,g)$ be a closed Riemannian manifold with negative sectional curvature, of dimension $n\geq 3$. An immersed totally geodesic submanifold $N$ in $M$ is maximal if it is not c…

Existence conjecture. There exist an orthogonal representation $\rho:\Gamma\to \operatorname{End}(H)$ weakly equivalent to the regular representation $\lambda_\Gamma$, and a…

Let $(M,g)$ be a compact connected Riemannian manifold of negative sectional curvature. A weak limit $\tilde\mu$ is a weak- limit of the probability measures…

Large-area rigidity conjecture. There exists $A=A(M)$ such that every minimal surface $\Sigma$ with area larger than $A$ is rigid.

Large totally geodesic hypersurface rigidity conjecture. For every $D>0$ and $k\in\mathbb{N}$, there exists $A=A(D,k)$ such that if $N$ contains a totally geodesic hyperbolic hyper…

Brin's conjecture. If $(M,g)$ is $\delta$-pinched for some $\delta>0.25$, then the frame flow is ergodic.

Let $M$ be a pinched Hadamard manifold, meaning a simply connected, complete Riemannian manifold whose sectional curvature is bounded between two negative constants. Let…

A closed Riemannian manifold $M$ has higher hyperbolic rank if every geodesic $c(t)$ in $M$ has a Jacobi field $J(t)$ such that $J(t)\neq 0$ and the sectional curvature of the plan…

Invariant-measure periodic-orbit conjecture. If $\mu(S^*H)>0$, then $\mu|_{S^*H}$ is supported on a union of periodic geodesics $\gamma$ such that $\gamma\cap S^*H$ has positive ar…

Let $M$ be a three-manifold equipped with a negatively curved metric, and let the Cross Curvature Flow (XCF) evolve this metric. A hyperbolic metric is a metric of constant negativ…

Let $M$ be a Riemannian manifold and let $N$ be a Riemannian manifold of negative sectional curvature. Consider the energy functional on a connected component of…

Negative curvature conjecture. The sectional curvature $k_\sigma^N$ is negative at every immersed point. Consequently, the induced curvature $\kappa$ is negative at every immersed…