Shapes of geodesic nets

Abstract

Let $M^n$ be a closed Riemannian manifold of dimension $n$. In this paper we will show that either the length of a shortest periodic geodesic on $M^n$ does not exceed $\left(n+1\right)d$, where $d$ is the diameter of $M^n$ or there exist infinitely many geometrically distinct stationary closed geodesic nets on this manifold. We will also show that either the length of a shortest periodic geodesic is, similarly, bounded in terms of the volume of a manifold $M^n$, or there exist infinitely many geometrically distinct stationary closed geodesic nets on $M^n$.