15 problems

Let $M$ be a compact $n$-dimensional Riemannian manifold, and let $\omega_p^1$ denote the Almgren–Pitts min-max value for $p$-sweepouts through 1-cycles. Gromov's Weyl law conjectu…

Nabutovsky's conjecture. There exist geodesic nets in the Euclidean plane with $4$ unbalanced vertices and an arbitrarily large number of balanced vertices. Moreover, this may alre…

A geodesic net in the Euclidean plane is a finite multigraph embedded in the plane whose edges are geodesic segments; its vertices are partitioned into balanced and unbalanced vert…

Let $M$ be a complete, non-compact Riemannian manifold with locally convex ends. Chambers–Liokumovich–Nabutovsky–Rotman conjecture. Every such manifold admits a closed geodesic. Th…

Let $(M,g)$ be a Riemannian manifold, and call two stationary geodesic nets distinct when they are not the same net. Infinitude conjecture. Every Riemannian manifold $(M,g)$ contai…

Let $M$ be a closed manifold, let $g$ be a Riemannian metric on $M$, and let $\gamma_i:\Gamma_i\rightarrow M$ be stationary geodesic nets. A sequence of such nets is equidistribute…

For the finite sequence of layer angles $\varphi_i(\varphi)$ in the construction $G_n(\varphi)$, take a sufficiently small nonzero parameter $\varphi\in(-\epsilon,\epsilon)$. Disti…

For the finite layered construction $G_n(\varphi)$, let $\varphi_i(\varphi)$ denote the suspension angle at layer $i$, and let $\varphi_i'(0)$ be its derivative at $\varphi=0$. Sus…

Let $M$ be a complete Riemannian manifold. For a geodesic net $G$ on $M$, let $L$ be its total length, $\widetilde L$ its adjusted length, $D$ its diameter, and $n$ the number of u…

Let a geodesic multinet in the Euclidean plane have $n$ unbalanced vertices, diameter $D$, total length $L$, and some number of balanced vertices. Length-diameter boundedness conje…

Consider geodesic nets for which all imbalances are equal to one. Mermarian's conjecture. In this case, the total imbalance equals the number of unbalanced vertices, and the number…

Let a geodesic net in the Euclidean plane have a finite number of unbalanced vertices, each with an imbalance, and let the total imbalance be the sum of those imbalances. Gromov's…

Let $N_0$ be a positive integer and let $S$ be an $N_0$-tuple of points in $\bigl(\mathbb{R}^2\bigr)^{N_0}$. Generic unboundedness conjecture. There exist $N_0$ and an $N_0$-tuple…

Let a geodesic net in the Euclidean plane have four unbalanced vertices and some number of balanced vertices. Four-vertex unboundedness conjecture. There exist geodesic nets in the…

Let $n$ be a positive integer, and let a smooth $2$-sphere be equipped with a smooth Riemannian metric. A geodesic net is a geodesic graph whose vertices have incident edges meetin…