80 problems

Let $M$ be a closed Riemannian manifold without conjugate points, and let the Liouville measure denote the natural invariant probability measure for its geodesic flow. Freire–Mañé'…

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

Entropy-volume conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$,

Density conjecture. $\mathcal M^\ddagger(M)$ is dense in the space of metrics of the gradient type.

Let $(M^n,g)$ be a pseudo-Riemannian manifold of dimension $n\ge 2$. A projective transformation is a diffeomorphism of $M^n$ that maps unparameterized geodesics to geodesics. Let…

Let a closed manifold carry a geodesic flow that is analytically integrable, meaning integrable by real-analytic first integrals in the sense used in the source, and let…

Let a Riemannian manifold carry an integrable geodesic flow, and denote the flow's topological entropy by $h_{\mathrm{top}}$ and the manifold's fundamental group by $\pi_1(M)$. A f…

Hyperbolicity conjecture. Any projective connection can be perturbed to a projective connection all of whose geodesics are hyperbolic, without altering its unparameterized geodesic…

Generic horseshoe conjecture. Generically in the $C^k$ topology, the geodesic flow of $g$ has a horseshoe.

Uniform boundedness conjecture. There is a constant $C$ depending only on the genus of $S$, the area of $S$, the minimum length of the geodesics in $S$, and the minimum of the curv…

Let $M$ be a manifold, let $\sigma$ be a symplectic form on $M$, and let $H$ be the twisted geodesic Hamiltonian on $T^*M$. Magnus conjecture. For every $M$ and any symplectic form…

Let $G$ be a compact Lie group, let $H$ be a subgroup, and let $\mathfrak v$ denote the corresponding tangent-space representation used to form the invariant polynomial algebra…

Let $T^2$ be the two-dimensional torus, and consider an integrable geodesic flow on its cotangent bundle. An integral quadratic in momenta is a first integral of the geodesic flow…

Let $(T^1M,f^t)$ be the geodesic flow on a compact rank $1$ surface of nonpositive curvature. For $\alpha\geq 0$, let $L(\alpha)$ be the level set of Lyapunov exponents, and write…

Let $g$ be a Riemannian metric whose geodesic flow is robustly transitive, meaning that $g$ admits a $C^2$-neighbourhood such that every metric in that neighbourhood has topologica…

Let $\mathcal{A}$ be the set of compact Riemannian surfaces whose geodesic flows are Anosov, and consider the $C^{2}$ topology on the space of Riemannian metrics. Let $\mathcal{N}$…

Let $\textbackslash Sigma\textbackslash subset\textbackslash mathbb R^d$ be a smooth bounded domain, and consider its periodic geodesics. The property in question is that the set o…

Let $g$ be a metric on the two-dimensional torus $\mathbb{T}^2$. A metric is polynomially integrable when its second conserved quantity is a polynomial in the momentum variables, a…

Folklore conjecture. Liouville metrics are the only integrable Riemannian metrics on $\mathbb{T}^2$. This is viewed as a closed-manifold analogue of the Birkhoff conjecture and is…

Zero-Liouville-measure conjecture. The set $\Lambda_0$ has zero Liouville measure.

Periodic-orbit Lyapunov rigidity conjecture. If the unstable Lyapunov exponents are constant across all periodic orbits, then $M$ has constant negative sectional curvature.

Florio–Hryniewicz conjecture. The good bound for $\delta$ should be $1/4$: among arbitrary positively curved $2$-spheres, the geodesic flow on the unit tangent bundle should be lef…

Liouville-metric conjecture. Liouville metrics are the only integrable metrics on tori, modulo the possible restriction to integrals that are polynomial in the momenta; flat metric…

Let the geodesic flow be the Hamiltonian flow of a Riemannian metric on the 2-torus. A polynomial first integral is a polynomial in the momenta that Poisson-commutes with the Hamil…

Let $X(\{\ell_n\},\{t_n\})$ be a flute surface whose cuff lengths form a non-decreasing sequence $\{\ell_n\}$, and let each twist parameter satisfy $t_n\in\{0,1/2\}$. Strong Kahn–M…