Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces

Abstract

We prove that if $M$ is a closed Riemannian surface of diameter $d$ and area $v$ with sectional curvature in the $[-1,1]$ interval, then a closed geodesic net of length $l$ has at most $f(l,d,v)$ branch points, where $f(l,d,v)=(400\tilde{l}{)}^{(180\tilde{l}{)}^4}$ for $\tilde{l}=\max \{l,\frac{\exp (d)}{\min \{1,\frac{v}{4}\}}\}$ This answers a question posed by S. Becker-Kahn. We also prove that for each geodesic net in the Euclidean plane with at most n unbalanced (boundary) vertices such that all its unbalanced vertices have degree 1, the number of balanced vertices of degree $\ge 3$ (=branch points) does not exceed $(25n{)}^{2n^2}$. This answers a question posed in [GM] and [NP].