Volume 11, issue 2 (2007)

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Shapes of geodesic nets

Alexander Nabutovsky and Regina Rotman

Geometry & Topology 11 (2007) 1225–1254
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 ${M}^{n}$.

Keywords
closed geodesics, geodesic nets, geometric calculus of variations
Mathematical Subject Classification 2000
Primary: 53C22, 53C23
Secondary: 58E10, 58E35