Geodesic nets with three boundary vertices

2018/03/10 by Parsch, Fabian · 1 citation
#Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1803.03728

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.

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• Towards Constructing Geodesic Nets with Four Boundary Vertices and an Increasing Number of Balanced Vertices

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