Shapes of geodesic nets

Nabutovsky, Alexander¹ ; Rotman, Regina¹

¹ Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada, Department of Mathematics, Penn State University, University Park PA 16802, USA

Geometry & topology, Tome 11 (2007) no. 2, pp. 1225-1254.

Résumé

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 $M^n$.

Détail

DOI : 10.2140/gt.2007.11.1225

Keywords: closed geodesics, geodesic nets, geometric calculus of variations

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