# Geodesic nets with three boundary vertices

## TL;DR

This paper proves a new geometric property of geodesic nets with three boundary vertices on non-positively curved planes, showing they can have at most one balanced vertex, with implications for Euclidean and curved geometries.

## Contribution

It establishes a novel bound on the number of balanced vertices in geodesic nets with three boundary points on non-positively curved surfaces, extending understanding in geometric network theory.

## Key findings

- At most one balanced vertex in such nets on non-positively curved planes
- The result does not hold for positively curved metrics
- No straightforward extension to four boundary vertices

## Abstract

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.