10 problems

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…

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…

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 $G=(B,U,E)$ be a geodesic net in the plane, with balanced vertices $B$ and unbalanced vertices $U$. For a vertex $v$, let $\operatorname{imb}(v)$ denote the norm of the sum of…