Riemannian metrics on the sphere with Zoll families of minimal hypersurfaces

TL;DR

This paper constructs special Riemannian metrics on spheres with smooth Zoll families of minimal hypersurfaces, extending Guillemin's theorem and addressing Yau's question on minimal hypersurface perturbations.

Contribution

It generalizes Guillemin's theorem to higher dimensions and minimal hypersurfaces, using Nash-Moser techniques to construct such metrics on spheres and projective spaces.

Findings

Existence of smooth Zoll metrics with minimal hypersurfaces on spheres

Characterization of metrics with minimal equators on projective spaces

Extension of Guillemin's geodesic results to higher-dimensional minimal hypersurfaces

Abstract

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.