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Equivariant constructions of spheres with Zoll families of minimal spheres

Lucas Ambrozio, Diego Guajardo

TL;DR

This work constructs, in every dimension $n\ge 3$, smooth one-parameter deformations of the Euclidean sphere $\mathbb{S}^n$ inside $\mathbb{R}^{n+1}$ that carry a Zoll family of codimension-one embedded minimal hypersurfaces, with the construction being $O(n+1)$-equivariant. The authors develop an equivariant Nash–Moser–type implicit-function approach, combining an Area operator, a Funk transform (and its dual), and a tame nonlinear scheme to produce the deformations $g_t$ with $g_t= i_t^*\mathrm{can}$ that admit a Zoll family parametrized by $\mathbb{RP}^n$. They extend these ideas to produce Zoll metrics on $\mathbb{RP}^n$ with minimal real projective hyperplanes not isometric to metrics with minimal linear hyperplanes, and show that any finite subgroup of $O(3)$ not containing $-\mathrm{Id}$ arises as the isometry group of some Zoll metric on $\mathbb{S}^2$. The paper also provides a detailed symmetry analysis, including Type I and III cases, and includes an appendix with explicit formulas for star-shaped embeddings, enabling precise comparisons with previous conformal constructions. Altogether, the results yield new Zoll-like geometries in all dimensions and offer a framework for exploiting symmetry in the study of minimal Zoll families.

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 $\mathbb{S}^2$.

Equivariant constructions of spheres with Zoll families of minimal spheres

TL;DR

This work constructs, in every dimension , smooth one-parameter deformations of the Euclidean sphere inside that carry a Zoll family of codimension-one embedded minimal hypersurfaces, with the construction being -equivariant. The authors develop an equivariant Nash–Moser–type implicit-function approach, combining an Area operator, a Funk transform (and its dual), and a tame nonlinear scheme to produce the deformations with that admit a Zoll family parametrized by . They extend these ideas to produce Zoll metrics on with minimal real projective hyperplanes not isometric to metrics with minimal linear hyperplanes, and show that any finite subgroup of not containing arises as the isometry group of some Zoll metric on . The paper also provides a detailed symmetry analysis, including Type I and III cases, and includes an appendix with explicit formulas for star-shaped embeddings, enabling precise comparisons with previous conformal constructions. Altogether, the results yield new Zoll-like geometries in all dimensions and offer a framework for exploiting symmetry in the study of minimal Zoll families.

Abstract

We construct one-parameter deformations of the Euclidean sphere inside that admit a Zoll family of codimension one embedded minimal spheres, in all dimensions . 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 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 , in all dimensions , 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 . As an application, we also show that every finite group of the orthogonal group that does not contain is the isometry group of some (classical) Zoll metric on .

Paper Structure

This paper contains 28 sections, 35 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Preliminaries and notations
  4. Construction
  5. The Area operator
  6. The Funk transform and its dual
  7. The Nash-Moser framework
  8. Symmetry
  9. Equivariance
  10. Applications
  11. Computation of the isometry group
  12. Zoll metrics on $\mathbb{S}^2$ with discrete group of isometries
  13. Symmetries of Type $III$
  14. $D_n[D_{2n}$-symmetry
  15. $\mathbb{Z}_n[D_n$-symmetry
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $f$ be a smooth odd function on $\mathbb{S}^n$, $n\geq 2$. Then there exists a smooth one-parameter family of functions $\psi_t\in C^{\infty}(\mathbb{S}^n)$, $|t|<\varepsilon$, with $\psi_t = tf + o(t)$ as $t$ goes to zero, such that is an embedding with $g_{t}=\iota_t^*can\in\mathcal{Z}$ for all $t\in (-\varepsilon,\varepsilon)$.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 74 more