A Zoll counterexample to a geodesic length conjecture

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.

💡 Research Summary

The paper addresses a long‑standing conjecture in Riemannian geometry concerning the relationship between two fundamental quantities on a closed surface: the diameter (D) (the maximal distance between any two points) and the length (L) of the shortest non‑trivial closed geodesic. The conjecture, motivated by the round sphere, asserts that for any Riemannian metric on the 2‑sphere one should have
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