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Stability of nearly Kähler and nearly parallel $G_2$-manifolds

Enric Solé-Farré

Abstract

We investigate generalisations of Hitchin's functionals, whose critical points correspond to nearly Kähler and nearly parallel $G_2$-structures. Our focus is on the gradient flow of these functionals and the spectral decomposition of their Hessians with respect to natural indefinite inner products. We introduce a Morse-like index for these functionals, termed the Hitchin index. We prove this index provides a lower bound for the Einstein co-index and explore its relationship with the deformation theory of $G_2$- and $\operatorname{Spin}(7)$-conifolds.

Stability of nearly Kähler and nearly parallel $G_2$-manifolds

Abstract

We investigate generalisations of Hitchin's functionals, whose critical points correspond to nearly Kähler and nearly parallel -structures. Our focus is on the gradient flow of these functionals and the spectral decomposition of their Hessians with respect to natural indefinite inner products. We introduce a Morse-like index for these functionals, termed the Hitchin index. We prove this index provides a lower bound for the Einstein co-index and explore its relationship with the deformation theory of - and -conifolds.

Paper Structure

This paper contains 13 sections, 65 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Stable forms and Hitchin functionals
  3. Nearly Kähler structures
  4. The nearly Kähler Hitchin functional
  5. The closed Hitchin functional
  6. Nearly parallel $G_2$-structure
  7. The nearly parallel $G_2$ Hitchin functional
  8. The new $G_2$ Hitchin functional
  9. Outlook
  10. Non parabolicity of the nearly Kähler Laplacian flow
  11. The DeTurck vector fields
  12. The nearly Kähler Laplacian flow
  13. The Einstein--Hilbert action

Key Result

Theorem 1

The new Hitchin functional $\mathcal{Q}: \mathcal{U}\rightarrow \mathbb{R}$ satisfies the following.

Theorems & Definitions (114)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Theorem 2.2: Hitchin00
  • Example 2.3: 6-dimensions Hitchin00
  • Example 2.4: 7-dimensions Hitchin00
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • ...and 104 more