Volume 11, issue 2 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Shapes of geodesic nets

Alexander Nabutovsky and Regina Rotman

Geometry & Topology 11 (2007) 1225–1254
Abstract

Let Mn be a closed Riemannian manifold of dimension n. In this paper we will show that either the length of a shortest periodic geodesic on Mn does not exceed (n + 1)d, where d is the diameter of Mn 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 Mn, or there exist infinitely many geometrically distinct stationary closed geodesic nets on Mn.

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