A Zoll Counterexample to a Geodesic Length Conjecture
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- Balacheff, F., Croke, C. & Katz, M.G. Geom. Funct. Anal. (2009) 19: 1. doi:10.1007/s00039-009-0708-9
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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.