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Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration

Athanasios Georgakopoulos, Marco Magliaro, Luciano Mari, Andreas Savas-Halilaj

TL;DR

This work addresses the rigidity of harmonic maps $f:\mathbb{S}^3\to\mathbb{S}^2$ by connecting Hessian bounds and singular-value data to a Hopf fibration framework. Using a singular-value-based analysis and Bochner-Weitzenböck formulas, the authors prove pinching-type rigidity results that force $f$ to factor through the Hopf fibration composed with conformal (or Möbius) maps when the Hessian and 2-dilation satisfy precise inequalities. A key feature is the use of maximum-principle arguments on combinations of the singular values, avoiding derivative estimates on the Hessian, to obtain Simons-type rigidity results. Additional characterizations show that under constant energy density or constant $D_2$, harmonic maps must be Hopf fibrations, highlighting the Hopf fibration as a central rigid object in this setting.

Abstract

It was conjectured by Eells that the only harmonic maps $f : S^3 \to S^2$ are Hopf fibrations composed with conformal maps of $S^2$. We support this conjecture by proving its validity under suitable conditions on the Hessian and the singular values of $f$. Among the results, we obtain a pinching theorem in the spirit of that of Simons, Lawson and Chern, do Carmo and Kobayashi for minimal hypersurfaces in the sphere.

Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration

TL;DR

This work addresses the rigidity of harmonic maps by connecting Hessian bounds and singular-value data to a Hopf fibration framework. Using a singular-value-based analysis and Bochner-Weitzenböck formulas, the authors prove pinching-type rigidity results that force to factor through the Hopf fibration composed with conformal (or Möbius) maps when the Hessian and 2-dilation satisfy precise inequalities. A key feature is the use of maximum-principle arguments on combinations of the singular values, avoiding derivative estimates on the Hessian, to obtain Simons-type rigidity results. Additional characterizations show that under constant energy density or constant , harmonic maps must be Hopf fibrations, highlighting the Hopf fibration as a central rigid object in this setting.

Abstract

It was conjectured by Eells that the only harmonic maps are Hopf fibrations composed with conformal maps of . We support this conjecture by proving its validity under suitable conditions on the Hessian and the singular values of . Among the results, we obtain a pinching theorem in the spirit of that of Simons, Lawson and Chern, do Carmo and Kobayashi for minimal hypersurfaces in the sphere.

Paper Structure

This paper contains 13 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Basics on maps from $\mathbb{S}^3$ to $\mathbb{S}^2$
  3. The singular value decomposition
  4. The Hessian and the tension field
  5. The Hopf fibration and harmonic unit vector fields
  6. The Hopf fibration
  7. Harmonic unit vector fields on $\mathbb{S}^3$
  8. Bochner-Weitzenböck formulas
  9. Weakly conformal harmonic maps from $\mathbb{S}^3$ to $\mathbb{S}^2$
  10. Proofs of the main theorems
  11. Proof of Theorem \ref{['thmA']}
  12. Proof of Theorem \ref{['thmC']}
  13. Proof of Theorem \ref{['thmB']}

Key Result

Theorem A

Suppose that $f: \mathbb{S}^3\to\mathbb{S}^2$ is a harmonic map. Then the following hold:

Theorems & Definitions (32)

  • Conjecture : Eells
  • Theorem A
  • Remark 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 22 more