6 problems

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

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…

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

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…