  In this paper we construct smooth Riemannian metrics on the sphere which
admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem
of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse
Function Theorem in the tame maps setting of Hamilton. This answers a question
of Yau on perturbations of minimal hypersurfaces in positive Ricci curvature.
We also consider the case of the projective space and characterize those
metrics on the sphere with minimal equators.
