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arXiv · 2506.07674

Systolic inequalities on the sphere from symplectic embeddings

Brayan Ferreira
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Abstract

We use properties of symplectic capacities that were recently defined by Hutchings to obtain upper bounds on the minimal action of Reeb orbits on fiberwise star-shaped hypersurfaces $\Sigma \subset T^*S^2$. In addition, we introduce the notion of a fiberwise $\beta$-balanced hypersurface $\Sigma \subset T^*S^2$ and establish upper bounds for the systole in terms of $\beta$ and geometric data, in the case of Riemannian metrics on $S^2$ satisfying this property. Finally, under the assumption of antipodal symmetry, we provide a non-sharp estimate of how fiberwise balanced a $\delta$-pinched metric is.

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Brayan Ferreira. 2025-06-09. Systolic inequalities on the sphere from symplectic embeddings. https://arxiv.org/abs/2506.07674

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