For fiberwise β-balanced metrics on the sphere, the shortest closed geodesic has squared length at most π/β times the area, and antipodally symmetric δ-pinched metrics are shown to be fiberwise balanced in an explicit but extremely weak range.
Entropy collapse versus entropy rigidity for Reeb and Finsler flows
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Systolic inequalities on the sphere from symplectic embeddings
For fiberwise β-balanced metrics on the sphere, the shortest closed geodesic has squared length at most π/β times the area, and antipodally symmetric δ-pinched metrics are shown to be fiberwise balanced in an explicit but extremely weak range.