158 problems

Let $(S^2,g)$ be a Riemannian $2$-sphere, and let $kappa>0$. A closed curve has constant geodesic curvature $kappa$ if its geodesic curvature is equal to $kappa$ at every point. Ar…

Let a convex smooth surface be a smooth convex surface in $\mathbb{R}^3$, and let a closed simple geodesic be a closed geodesic that is non-self-intersecting. Poincaré's conjecture…

Let $M$ be a closed, orientable, arithmetic, hyperbolic $3$-manifold. The Short Geodesic Conjecture. There is a constant $C>0$, independent of $M$, such that the length of the shor…

Let $L$ be the directed landscape, let $(t)=_{0,0;0,1}(t)$ be the directed-landscape geodesic location, and let $L(t)=L(0,0;(t),t)$ be its segment length. Let…

Let $P$ be a convex polyhedron. A simple closed geodesic is a simple closed geodesic on the surface of $P$, and a simple closed quasigeodesic through exactly one vertex is a simple…

Let $G_\alpha$, with $-1\leq\alpha\leq 1$, be the homogeneous Riemannian 3-manifold equipped with its canonical left-invariant metric. For a geodesic segment $\gamma$ of length $T$…

Let $(X,\tilde g)$ be the universal cover of a $d$-dimensional smooth Riemannian manifold $(M,g)$. A mediatrix is the locus equidistant from the reference point and one of its nont…

Cross-blocking conjecture. A compact Riemannian manifold $(M,g)$ has cross blocking if and only if it is isometric to a compact rank one symmetric space. In particular, it has both…

Finite-blocking flatness conjecture. If $(M,g)$) has finite blocking, then $g$ is a flat metric.

Local minimality conjecture. A geodesic is locally minimal if and only if

Newman's conjecture. For a large class of stationary measures ${\mu}$,

Let $\xi=\xi(d)$ denote the wandering exponent, defined so that $|y-x|^\xi$ gives the typical or largest order of deviation of a geodesic from the straight line segment…

Let $d\ge2$ and consider Euclidean first-passage percolation with passage times determined by the parameter $\alpha\in(1,\infty)$. A doubly infinite geodesic is a geodesic extendin…

Let $Q$ be the Poisson point set in Euclidean space and let $R_\infty$ be the associated minimal spanning forest, whose edges are formed from the limiting minimum spanning construc…

Let $V$ be a Kähler manifold, let $\mathcal{H}$ be the space of Kähler potentials, and let $\mathcal{H}_0$ denote its normalized subspace. For a normalized $\phi\in\mathcal{H}_0$,…

Let $T_m$ be the modular punctured torus, let $A_m=1.08...$ be the area of its associated unit ball, and let $N_L(T_m)$ denote the number of geodesic loops of length at most $L$ on…

Let $\mathfrak{M}_{3|reg}$ be the regular thermo-metric event manifold. In the numerically studied case, geodesics reaching the boundary at infinity, identified with the boundary o…

Let $H_n$ denote the maximal height of a geodesic from $0$ to $ne_1$, where $e_1$ is the first vector of the canonical basis of $\mathbb{R}^d$, and let $\xi(d)$ be a constant depen…

For $d\ge 2$, consider classical first-passage percolation on $\mathbb{Z}^d$ with an arbitrary passage-time distribution. A geodesic between two vertices is a path attaining the pa…

Let $\gamma$ be the limiting curve obtained as the $H^1([0,1],M)$-limit of a subsequence of minimising trajectories for the unconstrained optimisation problems described above. Hig…

Let $M$ be a smooth closed surface, and consider two point vortices on $M$ whose separation is infinitesimal and whose vorticities are equal in magnitude and opposite in sign. Kimu…

Consider the dynamical site-resampling setting for last-passage percolation, and let $E$ be the set of times at which non-trivial bigeodesics exist. A bigeodesic is a bi-infinite d…

Let $x,y\in{\mathbb S}^{n-1}$ and $v_x\in T_x{\mathbb S}^{n-1}$. Define $\gamma_x(t)=\exp_x(tv_x)$ and $\gamma_y(t)=\exp_y(tv_y)$, where $v_y$ is the parallel transport of $v_x$ al…

Direct-type criterion. The geodesic $\gamma(t)$ is of direct-type if and only if

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant probability measure…