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.

中文翻译：

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具有三个边界顶点的测地线网

我们证明了在非正曲面上具有三个边界（=不平衡）顶点的测地线网络最多具有一个平衡顶点。我们不假设不平衡顶点的度数有任何先验界限。即使在欧几里得的情况下，结果似乎也是新的。我们通过示例证明，对于平面上的正曲率度量，该结果是不正确的，并且对于具有四个不平衡顶点的测地线网络，该结果没有立即推广。