Equivariant constructions of spheres with Zoll families of minimal spheres

TL;DR

The paper constructs new equivariant deformations of spheres and projective spaces with Zoll families of minimal spheres or hyperplanes, extending classical results to higher dimensions and revealing new geometric structures.

Contribution

It introduces novel equivariant methods to produce Zoll families of minimal spheres and hyperplanes in all dimensions, including non-isometric metrics on real projective spaces.

Findings

Existence of Zoll families of minimal spheres in all dimensions n≥3.

Construction of metrics on RP^n with Zoll families of minimal projective hyperplanes.

Every finite subgroup of O(3) not containing -Id is an isometry group of some Zoll metric on S^2.

Abstract

We construct one-parameter deformations of the Euclidean sphere $\mathbb{S}^n$ inside $\mathbb{R}^{n+1}$ that admit a Zoll family of codimension one embedded minimal spheres, in all dimensions $n\geq 3$. The method of construction is equivariant with respect to the natural actions of the orthogonal group. In particular, we show that the original Zoll spheres of revolution in $\mathbb{R}^3$ have counterparts in the context of minimal surface theory, in all dimensions. We also describe the first examples of metrics on the real projective spaces $\mathbb{RP}^n$, in all dimensions $n \geq 3$, that admit a Zoll family of embedded minimal projective hyperplanes, and which are not isometric to metrics with minimal linear projective hyperplanes. The new constructions are underpinned by equivariant versions of Nash-Moser-Hamilton implicit function theorem, and yield new information even in dimension $n=2$. As an application, we also show that every finite group of the orthogonal group $O(3)$ that does not contain $-Id$ is the isometry group of some (classical) Zoll metric on $\mathbb{S}^2$.