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 $\ge 3.$ 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.

Keywords:

• Geodesic nets
• Riemannian manifolds
• 1-dimensional stationary varifolds

Acknowledgments

This research has been partially supported by A. Nabutovsky’s NSERC Discovery grant and F. Parsch’s Vanier Canada Graduate Scholarship. The paper has been partially written while A. Nabutovsky was visiting the Institute for Advanced Study in Spring, 2019. He wants to thank the IAS for its warm hospitality. The authors would like to thank Adam Stinchcombe for valuable help with variable precision arithmetic in MATLAB. The authors would like to thank Frank Morgan and an anonymous referee for their numerous comments that helped to improve the exposition.

Declaration of Interest

No potential conflict of interest was reported by the author(s).