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Global minimality of the Hopf map in the Faddeev-Skyrme model with large coupling constant

André Guerra, Xavier Lamy, Konstantinos Zemas

TL;DR

The paper proves that the Hopf map is the global minimizer of the perturbed Faddeev–Skyrme energy FS_ρ among maps S^3→S^2 with Hopf charge 1, for 0<ρ≤1, unique up to rigid motions. It achieves this by introducing a relaxed energy on closed 2-forms, exploiting a Hodge–Laplacian spectral decomposition to identify a minimal subspace E_{0,1}^+ that characterizes minimizers, and showing energy dominance reduces to a finite-dimensional stability problem. The key technique connects Hopf invariant theory, gauge lifting, and conformality properties to obtain a rigorous rigidity result: any minimizer with the same Hopf data must be of the form u=h∘R with R∈SO(4). The results sharpen the understanding of energy-minimizing topological solitons in the Faddeev–Skyrme model and demonstrate a robust method to obtain uniqueness after relaxing the variational problem.

Abstract

We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev--Skyrme energy in its homotopy class, provided that the radius of the target 2-sphere is not smaller than the radius of the domain 3-sphere.

Global minimality of the Hopf map in the Faddeev-Skyrme model with large coupling constant

TL;DR

The paper proves that the Hopf map is the global minimizer of the perturbed Faddeev–Skyrme energy FS_ρ among maps S^3→S^2 with Hopf charge 1, for 0<ρ≤1, unique up to rigid motions. It achieves this by introducing a relaxed energy on closed 2-forms, exploiting a Hodge–Laplacian spectral decomposition to identify a minimal subspace E_{0,1}^+ that characterizes minimizers, and showing energy dominance reduces to a finite-dimensional stability problem. The key technique connects Hopf invariant theory, gauge lifting, and conformality properties to obtain a rigorous rigidity result: any minimizer with the same Hopf data must be of the form u=h∘R with R∈SO(4). The results sharpen the understanding of energy-minimizing topological solitons in the Faddeev–Skyrme model and demonstrate a robust method to obtain uniqueness after relaxing the variational problem.

Abstract

We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev--Skyrme energy in its homotopy class, provided that the radius of the target 2-sphere is not smaller than the radius of the domain 3-sphere.

Paper Structure

This paper contains 8 sections, 19 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some basic identities involving differential forms
  4. Spectral properties of the Hodge-Laplacian on closed forms
  5. The Hopf map and horizontal conformality
  6. The Hopf invariant for maps of finite Faddeev--Skyrme energy
  7. Relaxed form of the perturbed Faddeev--Skyrme energy
  8. Proof of uniqueness in Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

For every $\rho\in (0,\color{black} 1 ]\color{black}$ and $u\in \mathcal{U}_{\mathcal{FS}}$ with Hopf invariant equal to 1, it holds that with equality if and only if $u=h\circ R$ for some $R\in \textup{SO}(4)$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: Spectrum of $d\, *$
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 3.1
  • Proposition 3.2
  • ...and 28 more