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$G_2$-Poisson equation on homogeneous spheres

Stepan Hudecek

TL;DR

This work analyzes the Poisson equation for the non-linear $G_2$-Laplacian on invariant $3$-forms on the 7-sphere under transitive group actions. It classifies $G$-invariant $G_2$-structures for $G=Spin(7), SU(4), (Sp(2)\times Sp(1))/\mathbb{Z}_2$ and proves that the $G_2$-Laplacian is an orientation-preserving isomorphism on positive invariant $3$-forms, yielding existence and uniqueness of invariant solutions to $\Delta_{\varphi}\varphi=\mu$ for positive right-hand sides in these cases. For $G=(Sp(2)\times U(1))/\mathbb{Z}_2$, the operator may map out of the positive cone, and the paper provides a local bijection near the $(Sp(2)\times Sp(1))/\mathbb{Z}_2$-invariant locus, together with numerical evidence of non-uniqueness and a rich eigenstructure including nearly parallel $G_2$-structures. The results illuminate the landscape of invariant $G_2$-structures on homogeneous spheres and clarify how the nonlinearity of the $G_2$-Laplacian interacts with symmetry and positivity constraints. These findings have implications for constructing homogeneous $G_2$-manifolds and for understanding $G_2$-Laplacian dynamics on highly symmetric spaces.

Abstract

This paper studies the Poisson equation for the $G_2$-Laplacian on 3-forms on the 7-sphere that are invariant under a transitive group action. We establish the existence and uniqueness of $G$-invariant solutions for $G=SU(4),\: Spin(7),\: (Sp(2)\times Sp(1))/\mathbb{Z}_2$. In the case $G=Sp(2)\times U(1)/\mathbb{Z}_2$, we show that the operator does not preserve the set of positive 3-forms. The paper also discusses the eigenvalue problem for the $G_2$-Laplacian. We classify $G$-invariant solutions for the above choices of $G$ and determine which of these solutions are nearly parallel $G_2$-structures.

$G_2$-Poisson equation on homogeneous spheres

TL;DR

This work analyzes the Poisson equation for the non-linear -Laplacian on invariant -forms on the 7-sphere under transitive group actions. It classifies -invariant -structures for and proves that the -Laplacian is an orientation-preserving isomorphism on positive invariant -forms, yielding existence and uniqueness of invariant solutions to for positive right-hand sides in these cases. For , the operator may map out of the positive cone, and the paper provides a local bijection near the -invariant locus, together with numerical evidence of non-uniqueness and a rich eigenstructure including nearly parallel -structures. The results illuminate the landscape of invariant -structures on homogeneous spheres and clarify how the nonlinearity of the -Laplacian interacts with symmetry and positivity constraints. These findings have implications for constructing homogeneous -manifolds and for understanding -Laplacian dynamics on highly symmetric spaces.

Abstract

This paper studies the Poisson equation for the -Laplacian on 3-forms on the 7-sphere that are invariant under a transitive group action. We establish the existence and uniqueness of -invariant solutions for . In the case , we show that the operator does not preserve the set of positive 3-forms. The paper also discusses the eigenvalue problem for the -Laplacian. We classify -invariant solutions for the above choices of and determine which of these solutions are nearly parallel -structures.

Paper Structure

This paper contains 9 sections, 17 theorems, 114 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous $G_2$-structures
  4. Invariant $G_2$-structures on the sphere $S^7$
  5. Solving the Poisson equation
  6. $Spin(7)$-symmetry
  7. $SU(4)$-symmetry
  8. $Sp(2)\cdot Sp(1)$- and $Sp(2)\cdot U(1)$-symmetry
  9. Numerics for $Sp(2)\cdot U(1)$-symmetry

Key Result

Theorem 1.1

Let $G=SU(4),\: Spin(7)$ or $(Sp(2)\times Sp(1))/\mathbb{Z}_2$. Then the $G_2$-Laplacian is an orientation preserving isomorphism on the space of positive $G$-invariant 3-forms on the 7-sphere.

Figures (4)

  • Figure 1: Poset diagram of subgroups of $SO(8)$ acting transitively on $S^7$
  • Figure 2: Image of the $Sp(2)\cdot U(1)$-invariant rays from the origin in the plane given by $A=1,\alpha\in\{0,\pi\}$ under the $G_2$-Laplacian
  • Figure 3: Progressively bigger point clouds of the $(Sp(2)\cdot U(1))$-invariant forms mapped by the $G_2$-Laplacian
  • Figure 4: Point cloud of the $(Sp(2)\cdot U(1))$-invariant forms mapped by the $G_2$-Laplacian from the 'side'

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 30 more