Shapes of geodesic nets

Nabutovsky, Alexander; Rotman, Regina

doi:10.2140/gt.2007.11.1225

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Shapes of geodesic nets

Alexander Nabutovsky and Regina Rotman

Geometry & Topology 11 (2007) 1225–1254

DOI: 10.2140/gt.2007.11.1225

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 $(n + 1) d$ , 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

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Publication

Received: 17 February 2007

Revised: 20 April 2007

Accepted: 20 April 2007

Published: 20 June 2007

Proposed: Tobias Colding

Seconded: Leonid Polterovich, Jean-Pierre Otal

Authors

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

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

Volume 11, issue 2 (2007)