DOI: 10.1080/10586458.2020.1743216 URL: Authors: Alexander Nabutovsky, Fabian Parsch Journal/publisher: Taylor and Francis Inc. Year: 2020
Title: Geodesic nets: Some examples and open problems
Abstract: Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. In the first part of this paper, we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. Many of these open questions are about geodesic nets with edges of multiplicity one on the Euclidean plane. Our main focus is on relationships between the number of boundary vertices, the number of inner (or balanced) vertices, and some basic geometric characteristic of geodesic nets (such as the length or the imbalances at boundary vertices). The second part contains a new construction providing a partial answer for one of these questions: We describe an infinite family of geodesic nets with edges of multiplicity one on the Euclidean plane with a constant number (namely, 14) of boundary vertices and arbitrarily many inner (or balanced) vertices of degree (Formula presented.) The fact that all edges of the constructed geodesic nets have multiplicity one is not proven but strongly supported by numerical evidence obtained from experimentation.