Systolic inequalities on the sphere from symplectic embeddings

TL;DR

This paper applies symplectic capacity techniques to derive bounds on Reeb orbits and systolic inequalities on the sphere, introducing fiberwise balanced hypersurfaces and analyzing their geometric properties.

Contribution

It introduces the concept of fiberwise β-balanced hypersurfaces and establishes new upper bounds for systolic invariants using symplectic capacities.

Findings

Upper bounds on minimal Reeb orbit action using symplectic capacities

Definition and analysis of fiberwise β-balanced hypersurfaces

Estimate of fiberwise balanced property for δ-pinched metrics

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.