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Explicit solutions to the gradient flow of Spin(7)-structures

Joseph Duthie

Abstract

We study the gradient flow of Spin($7$)-structures and construct the first explicit solutions, in the homogeneous setting. As an intermediate step, we obtain formulae expressing the Spin($7$)-torsion tensor and gradient flow in terms of the Spin($7$)-torsion forms, which makes explicit computations more tractable. We use these formulae to find explicit solutions to the gradient flow of Spin($7$)-structures, obtaining a shrinking soliton on $\mathrm{SU}(3)$ as well as another explicit solution on a certain $T^7$-bundle over $S^1$. We also find an explicit solution to the coupled Ricci-harmonic flow of Spin($7$)-structures. Finally, we consider the question of stability of solitons for the renormalised gradient flow, and show that the soliton on $\mathrm{SU}(3)$ admits stable directions, unstable directions, and zero modes.

Explicit solutions to the gradient flow of Spin(7)-structures

Abstract

We study the gradient flow of Spin()-structures and construct the first explicit solutions, in the homogeneous setting. As an intermediate step, we obtain formulae expressing the Spin()-torsion tensor and gradient flow in terms of the Spin()-torsion forms, which makes explicit computations more tractable. We use these formulae to find explicit solutions to the gradient flow of Spin()-structures, obtaining a shrinking soliton on as well as another explicit solution on a certain -bundle over . We also find an explicit solution to the coupled Ricci-harmonic flow of Spin()-structures. Finally, we consider the question of stability of solitons for the renormalised gradient flow, and show that the soliton on admits stable directions, unstable directions, and zero modes.

Paper Structure

This paper contains 10 sections, 23 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Spin(7)-structures
  3. Decomposition of the space of forms
  4. The torsion tensor and torsion forms of a Spin(7)-structure
  5. A gradient flow of Spin(7)-structures
  6. Other flows of Spin(7)-structures
  7. A reformulation of the gradient flow
  8. Explicit solutions on homogeneous manifolds
  9. Linear instability of solitons
  10. Computational details for the gradient flow on SU(3)

Key Result

Theorem 1

Writing the gradient flow of Spin$(7)$-structures GF as we can express $A$ as where $\delta = -*_{\Phi_t}d*_{\Phi_t}$ is the codifferential of forms, $T^1_8$ and $T^5_{48}$ are the torsion forms, $\pi_7$ is the projection $\pi_7: \Omega^2 \rightarrow\Omega^2_7$ and $\mathsf{j}$ is a map from symmetric $2$-tensors to $4$-forms, depending on $\Phi_t$ and defined in Section Sect

Theorems & Definitions (45)

  • Definition 1.1: Dwivedi24
  • Theorem : \ref{['ThmTorsionForms']}
  • Theorem 1.2
  • Theorem : \ref{['ExampleSU3Unstable']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: LawsonMichelson
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6: Bonan
  • ...and 35 more