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.
"Geodesic nets with three boundary vertices." J. Differential Geom. 119 (1) 99 - 140, September 2021. https://doi.org/10.4310/jdg/1631124286