Equivariant constructions of spheres with Zoll families of minimal spheres

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 $n\ge 3$. 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 $n\ge 3$, 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 ${\mathbb{S}}^2$.

Reviewed version. To appear in Advances in Mathematics. 48 pages