# Geodesic Nets: Some Examples and Open Problems

## TL;DR

This paper surveys known results and open problems about geodesic nets on Riemannian manifolds, especially the Euclidean plane, and provides a new infinite family of such nets with specific boundary and interior vertices.

## Contribution

It offers a survey of open questions on geodesic nets and introduces a novel infinite family of geodesic nets with prescribed boundary and interior vertices on the Euclidean plane.

## Key findings

- Survey of open problems on geodesic nets
- Introduction of a new infinite family of geodesic nets
- Partial answer to a specific open question

## 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 half 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 on the Euclidean plane. The second half contains a partial answer for one of these questions, namely, a description of a new infinite family of geodesic nets on the Euclidean plane with 14 boundary (or unbalanced) vertices and arbitrarily many inner (or balanced) vertices of degree $\geq 3$.