Geometric and Functional Analysis

, Volume 19, Issue 1, pp 1-10

First online:

A Zoll Counterexample to a Geodesic Length Conjecture

  • Florent BalacheffAffiliated withSection de Mathématiques, Université de Genève
  • , Christopher CrokeAffiliated withDepartment of Mathematics, University of Pennsylvania
  • , Mikhail G. KatzAffiliated withDepartment of Mathematics, Bar Ilan University Email author 

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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.

Keywords and phrases:

Closed geodesic diameter Guillemin deformation sphere systole Zoll surface

AMS Mathematics Subject Classification:

53C23 53C22