Geodesic nets with three boundary vertices (Q2073256)

In any surface with a Riemannian metric, a \textit{geodesic net} is a connected graph, embedded in the surface, with geodesic edges. A vertex is \textit{balanced} if the sum of outgoing unit tangents to its adjacent edges vanishes. Example: In the Euclidean plane, a triangle is a geodesic net whose vertices are all unbalanced. Example: given any three points of the Euclidean plane, forming the vertices of an acute triangle, we can find a fourth point so that the line segments from it to the three balance. The author proves that this configuration has the maximum number of balanced vertices with the given unbalanced vertices. He proves indeed that, in any nowhere positively curved surface, any geodesic net with three unbalanced vertices has at most one balanced vertex. He draws counterexamples with positively curved metrics, and shows that with four unbalanced vertices there is no straightforward generalization. The method of proof is very geometric and accessible and does not use heavy equipment.