Geometric and Functional Analysis

, Volume 19, Issue 1, pp 1–10

A Zoll Counterexample to a Geodesic Length Conjecture

  • Florent Balacheff
  • Christopher Croke
  • Mikhail G. Katz

DOI: 10.1007/s00039-009-0708-9

Cite this article as:
Balacheff, F., Croke, C. & Katz, M.G. Geom. Funct. Anal. (2009) 19: 1. doi:10.1007/s00039-009-0708-9


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 geodesicdiameterGuillemin deformationspheresystoleZoll surface

AMS Mathematics Subject Classification:


Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  • Florent Balacheff
    • 1
  • Christopher Croke
    • 2
  • Mikhail G. Katz
    • 3
  1. 1.Section de MathématiquesUniversité de GenèveGenèveSuisse
  2. 2.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  3. 3.Department of MathematicsBar Ilan UniversityRamat GanIsrael