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Harmonic flow of $\mathrm{Spin}(7)$-structures

Shubham Dwivedi, Eric Loubeau, Henrique N. Sá Earp

Abstract

We formulate and study the isometric flow of $\mathrm{Spin}(7)$-structures on compact $8$-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of $H$-structures. We then specialise to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an $\varepsilon$-regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-$\mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric $\mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of independent interest.

Harmonic flow of $\mathrm{Spin}(7)$-structures

Abstract

We formulate and study the isometric flow of -structures on compact -manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of -structures. We then specialise to , obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an -regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type- singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric -structures, based on squares of spinors, which may be of independent interest.

Paper Structure

This paper contains 35 sections, 41 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. $H$-Structures, homogeneous sections and harmonic flow
  3. $H$-Structures
  4. The geometric setting of $H$-structures
  5. Deformations of $H$-structures
  6. Harmonic $H$-solitons and self-similar solutions
  7. Harmonic sections
  8. Vertical second fundamental form
  9. Weitzenböck and Bochner formulas
  10. Shi-type estimates along \ref{['eq: HSF']}
  11. Preliminaries on $\mathrm{Spin}(7)$-structures
  12. Definitions and basic properties
  13. Decomposition of the space of forms
  14. Proof of Theorem \ref{['prop: Spin(7) Bryant formula']}
  15. Harmonic $\mathrm{Spin}(7)$-flow
  16. ...and 20 more sections

Key Result

Theorem A

Let $(M^8,\Phi_0,g_0)$ be a $\mathrm{Spin}(7)$-structure manifold. Then any other $\mathrm{Spin}(7)$-structure inducing the same Riemannian metric $g_0$ can be parametrised by a function $f\in C^\infty(M)$ and a vector field $X\in\mathscr{X}(M)$ such that $f^2 + |X|^2 = 1$: where the $4$-form $\Theta_{0,X}(a,b,c,d) := \Phi_0 (a \cdot X,b,c,d)$ is defined by spinorial multiplication.

Theorems & Definitions (85)

  • Theorem A
  • Definition 1.1: Harmonic $\mathrm{Spin}(7)$-flow
  • Theorem B
  • Theorem C: small torsion and entropy
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 75 more