Systolic inequalities on the sphere from symplectic embeddings

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.