Geometric and Functional Analysis

, Volume 19, Issue 1, pp 1–10

A Zoll Counterexample to a Geodesic Length Conjecture

Authors

  • Florent Balacheff
    • Section de MathématiquesUniversité de Genève
  • Christopher Croke
    • Department of MathematicsUniversity of Pennsylvania
    • Department of MathematicsBar Ilan University
Article

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

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.

Keywords and phrases:

Closed geodesic diameter Guillemin deformation sphere systole Zoll surface

AMS Mathematics Subject Classification:

53C23 53C22

Copyright information

© Birkhäuser Verlag, Basel 2009