Uniqueness of hypersurface Zoll families

Prove uniqueness of hypersurface Zoll families in all dimensions for a given hypersurface Zoll metric.

Background

In dimension three, the paper states a uniqueness result for the surface Zoll family associated with a surface Zoll metric. It then explains that the analogous assertion in higher dimensions is not established; even the canonical metric on the four-sphere admits minimal three-spheres that are not equators, illustrating the obstruction to identifying all relevant minimal hypersurfaces with a single distinguished family.