11 problems

Security conjecture. A closed Riemannian manifold is secure if and only if it is flat.

A Finsler metric $F$ on the sphere $S^d$ is an irreversible Finsler metric defining the geodesic flow under consideration, and distinct closed geodesics are counted as geometricall…

The Weil–Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of Riemann surfaces, and WP geodesic rays have ending laminations. Ending-lamination parametri…

Entropy-or-flatness conjecture. Either $M$ is flat or its geodesic flow has positive topological entropy.

The geodesic curves of the random fields dynamics are considered on a parametric space whose curvature may vary along their trajectories; the trajectories are described by the Eule…

Let $M$ be a compact simply connected manifold satisfying … where $x$ has degree $d\geq 2$, height $n+1\geq 2$, and $\dim M=dn$. Let $F$ be any irreversible Finsler metric on $M$,…

Abate and Tovena's conjecture. There is a neighborhood $U$ of $p_0$ such that every geodesic ray which enters into $U$ tends to $p_0$. The two rays of any geodesic which stays in…

Let $S^o=\mathbb{P}^1(\mathbb{C})\setminus\{p_0,p_1,\dots,p_r\}$, where $p_0,p_1,\dots,p_r$ are the poles of a meromorphic connection $\nabla$ on $\mathbb{P}^1(\mathbb{C})$. Let…

Positive geodesic and hyperbolic-direction conjecture. The answers to these three questions should be positive, with greater reserve concerning the assertion that the set of hyperb…

Asymptotic growth conjecture.

Let $(M,g)$ be a closed $C^{\infty}$ Riemannian manifold. Here, $(M,g)$ is secure if every pair of points has the finite blocking property, and uniformly secure if there is a unifo…