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 by Balacheff, Croke, and Katz provides a definitive counterexample to the conjectured inequality L ≤ 2D for closed geodesics on the 2‑sphere, where L denotes the length of the shortest non‑trivial closed geodesic and D the diameter of the Riemannian metric. The conjecture, originally proposed by Nabutovsky and Rotman, suggested that the round metric (with L = 2π and D = π) should be optimal for the ratio L/D, and that no metric could achieve L > 2D.

The authors exploit the theory of Zoll surfaces, i.e., metrics for which every geodesic is closed and has the same length. A classical result of Guillemin (1976) states that for any smooth odd function f on S² there exists a one‑parameter family of Zoll metrics g_t = Ψ_f(t)·g₀, where g₀ is the round metric, Ψ_f(0)=1, and the infinitesimal conformal factor satisfies (dΨ_f/dt)_{t=0}=f. All geodesics of g_t retain length 2π, so L(g_t)=2π for every t.

The key innovation is the introduction of “amply negative” odd functions. An odd function f is called amply negative if, for every unit tangent vector (p,v) on S², there exists a great half‑circle τ issuing from p, forming an acute angle with v, such that the integral ∫_τ f ds₀ is uniformly bounded above by a negative constant –ν(f). The set of directions for which this holds must be “Y‑like”, meaning that its intersection with each unit circle contains a triple of vectors whose positive linear combination can vanish (equivalently, every open semicircle contains a direction from the set). This condition guarantees a rich supply of half‑circles on which f integrates negatively.

Section 2 proves that if f is amply negative then for any geodesic segment γ of length close to π there exists a piecewise geodesic τ (belonging to a specific family S P_γ) with ∫_τ f ds₀ < −ν(f). Lemma 2.3 establishes the existence of such τ uniformly for all near‑π segments.

Section 3 analyzes the energy functional E_t(τ)=∫|τ’|{g_t}² ds for a fixed path τ. Lemma 3.1 shows dE_t/dt|{t=0}=∫_τ f ds₀. Hence, for amply negative f, the energy (and therefore the length) of τ strictly decreases for small positive t. By selecting τ that connects any pair of points whose round‑metric distance is ≥π−ε, the authors obtain a new path whose g_t‑length is <π. Consequently, the diameter D(g_t) becomes strictly less than π for all sufficiently small t>0, while L(g_t)=2π remains unchanged. This yields L(g_t) > 2 D(g_t), violating the conjectured bound.

The remaining challenge is to exhibit an amply negative odd function. The authors construct such functions using “fine sets” X⊂S². A fine set satisfies three combinatorial conditions: (1) no three points are collinear, (2) no three great circles determined by pairs of points are concurrent away from X∪(−X), and (3) every open hemisphere contains at least three points of X. Fine sets exist; for example, a regular tetrahedron’s vertices can be perturbed to meet the conditions.

For a fine set X, they define smooth bump functions δ_ε(p) supported in small balls B(p,ε). The odd function f_ε = Σ_{p∈X} (δ_ε(−p) − δ_ε(p)) approximates a signed sum of Dirac deltas at the points of X and their antipodes. Lemma 5.2 shows that for any geodesic segment inside a ball, the integral of δ_ε is maximized by the diameter, so the signed sum yields negative integrals on a large family of half‑circles. Lemma 5.4 proves that for each unit tangent direction there exists a half‑circle τ of length π, making an acute angle with the direction, such that ∫_τ f_ε ds₀ < 0. Moreover, the collection of such directions is Y‑like, establishing that f_ε is amply negative for sufficiently small ε.

Putting all pieces together, the authors obtain a smooth odd function f that is amply negative, feed it into Guillemin’s deformation, and obtain a Zoll metric g_t with L(g_t)=2π and D(g_t)<π. This provides the first explicit counterexample to the L ≤ 2D conjecture on S². It also disproves a related conjecture of Nabutovsky–Rotman stating that every point of a closed Riemannian manifold should admit a closed geodesic loop of length ≤ 2D. In the constructed Zoll metrics, the shortest geodesic loop at any point has length 2π, while the diameter is strictly smaller than π.

Thus, the round sphere is not optimal for the ratio L/D, and the conjectured universal bound L ≤ 2D fails even in the highly symmetric setting of Zoll surfaces. The paper combines sophisticated tools from global Riemannian geometry, integral geometry, and careful combinatorial constructions to settle a long‑standing question.