77 problems

Rudnick–Sarnak conjecture. If $M$ is a compact Riemannian manifold of negative curvature, then every orthonormal basis of Laplacian eigenfunctions on $M$ should be quantum unique e…

Let $(X,g)$ be a closed, $2n$-dimensional Riemannian manifold with negative sectional curvature $\mathrm{sec}_{g}$. Let $h^k_{(2)}(X)$ denote the $k$-th $L^2$-Betti number of $X$.…

Let $M$ be a smooth closed manifold, and let $g$ and $g_0$ be negatively curved metrics, or more generally metrics whose geodesic flows are Anosov. Let $\mathcal L_g$ denote the ma…

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…

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

Let $(M,g)$ be a negatively curved Riemannian manifold, and let $FM$ be its frame bundle. The frame flow is the flow on $FM$ generated by the geodesic-frame vector field, and ergod…

Let $(X,g)$ be a smooth compact Riemannian surface of negative curvature, and let $\psi_\lambda\in L^2$ satisfy … Here $\\|\psi_\lambda\\|_{L^\infty}$ and…

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…

Let $(M,g)$ be a closed Riemannian manifold with all sectional curvatures negative, let $\Gamma=\pi_1(M)$, and let $C_\Gamma$ be the set of conjugacy classes of non-trivial element…

Let $f:M\rightarrow N$ be a homotopy equivalence between closed negatively curved manifolds. The unique harmonic map $h$ homotopic to $f$ is thereby defined. Lawson–Yau conjecture.…

Hopf-Chern conjecture.

Katok's Margulis-function conjecture. Unless $M$ is locally symmetric, $c(x)$ is almost always neither constant nor smooth.

Let $M$ be a negatively curved compact Kähler manifold of complex dimension at least $2$. Yau's rigidity conjecture. The complex structure of $M$ is unique. This conjecture general…

Negative curvature conjecture. The minimal immersion is never tangential to any flat inside the symmetric space. Consequently, the sectional curvature of the immersed minimal surfa…

Let $M$ be a negatively curved Riemannian manifold whose sectional curvature satisfies … Here, strict 1/4-pinching means that the sectional curvature lies strictly between $-1$ and…

Frame-flow ergodicity conjecture. The frame flow on $M$ should be ergodic. Frame-flow ergodicity is known in several settings, including constant curvature, generic negatively curv…

Conjecture on compact negatively curved space forms. Such a compact manifold exists if and only if $(p,q)$ occurs in the list

Let $R(\lambda)$ denote the remainder in the integrated Weyl law for the eigenvalue counting function on a surface of constant negative curvature. Randol's conjecture. For every…

For every integer $m>10$, there is a harmonic cellular map $h:M_1\rightarrow M_2$ between a pair of closed negatively curved $m$-dimensional Riemannian manifolds that is not a diff…

Let $M$ and $N$ be closed negatively curved Riemannian manifolds, and let $h:M\to N$ be a harmonic map that is not a diffeomorphism but is the limit of a 1-parameter family of diff…

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…

Reverse Riesz estimate. Then

Cross curvature flow conjecture. The cross curvature flow exists for all time and converges to a hyperbolic metric.

An almost flat manifold is a closed manifold admitting metrics of arbitrarily small sectional curvature and uniformly bounded diameter; a flat manifold is a closed manifold with a…