, Volume 19, Issue 1, pp 1-10

A Zoll Counterexample to a Geodesic Length Conjecture

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Abstract.

We construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin’s theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D.

Received: September 2007, Revision: November 2007, Accepted: November 2007
F.B. supported by the Swiss National Science Foundation. C.C. supported by NSF grants DMS 02-02536 and DMS 07-04145. M.K. supported by the Israel Science Foundation (grants 84/03 and 1294/06)