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A $G_2$-Hilbert functional in $G_2$-geometry

Panagiotis Gianniotis, George Zacharopoulos

TL;DR

This work introduces the $G_2$-Hilbert functional ${\mathcal F}$ on $G_2$-structures and shows it is uniquely characterized by being linear in scalar curvature and quadratic in torsion. The volume-normalized functional has torsion-free and nearly $G_2$-structures as saddle critical points, motivating two Ricci-flow–like evolutions $\partial_t\varphi=\hat P(\varphi)$ and $\partial_t\varphi=\tilde P(\varphi)$, built from special Ricci-like operators. The authors develop a thorough variational framework: they establish a Bianchi-type identity, derive explicit first and second variations, and decompose the tangent space into conformal, transverse-traceless, and diffeomorphism directions via the conformal Killing operator. In particular, the second variation is positive in conformal directions but possesses infinitely many negative TT directions, implying saddle behavior for static $G_2$-structures; conformal deformations of torsion-free or nearly $G_2$-structures are locally minimizing in their class. Altogether, the paper provides a variational foundation and analytic tools for $G_2$-flows that generalize Ricci-flow–type dynamics in $7$-manifolds with $G_2$-structure.

Abstract

In this paper we introduce a new functional on the space of $G_2$-structures which we call the $G_2$-Hilbert functional. It is uniquely determined by a few basic principles inspired by the Einstein-Hilbert functional in Riemannian Geometry, and it has similar variational behaviour with it. For instance, torsion-free and nearly $G_2$-structures are saddle critical points of the volume-normalized $G_2$-Hilbert functional. This allows us to uniquely distinguish two new flows of $G_2$-structures, which can be considered as analogues of the Ricci flow in $G_2$-geometry.

A $G_2$-Hilbert functional in $G_2$-geometry

TL;DR

This work introduces the -Hilbert functional on -structures and shows it is uniquely characterized by being linear in scalar curvature and quadratic in torsion. The volume-normalized functional has torsion-free and nearly -structures as saddle critical points, motivating two Ricci-flow–like evolutions and , built from special Ricci-like operators. The authors develop a thorough variational framework: they establish a Bianchi-type identity, derive explicit first and second variations, and decompose the tangent space into conformal, transverse-traceless, and diffeomorphism directions via the conformal Killing operator. In particular, the second variation is positive in conformal directions but possesses infinitely many negative TT directions, implying saddle behavior for static -structures; conformal deformations of torsion-free or nearly -structures are locally minimizing in their class. Altogether, the paper provides a variational foundation and analytic tools for -flows that generalize Ricci-flow–type dynamics in -manifolds with -structure.

Abstract

In this paper we introduce a new functional on the space of -structures which we call the -Hilbert functional. It is uniquely determined by a few basic principles inspired by the Einstein-Hilbert functional in Riemannian Geometry, and it has similar variational behaviour with it. For instance, torsion-free and nearly -structures are saddle critical points of the volume-normalized -Hilbert functional. This allows us to uniquely distinguish two new flows of -structures, which can be considered as analogues of the Ricci flow in -geometry.
Paper Structure (24 sections, 26 theorems, 260 equations)

This paper contains 24 sections, 26 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Preliminaries in $G_2$-geometry
  3. Contraction identities
  4. Decomposition of forms
  5. Torsion and the $G_2$-Bianchi identity
  6. Variations of $G_2$-structures
  7. Quasilinear second order differential operators on $G_2$-structures.
  8. Functionals in $G_2$-geometry
  9. Basic functionals in $G_2$-geometry
  10. Diffeomorphism invariance
  11. Special Ricci-like operators and diffeomorphism invariance
  12. The $G_2$-Hilbert functional
  13. Uniqueness of the $G_2$-Hilbert functional
  14. Bianchi-type identity
  15. Alternative expressions of the $G_2$-Hilbert functional
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $M$ be a compact $7$-manifold with a $G_2$-structure $\varphi_0$. Consider the initial value problem where $\mathrm{Ric}$ is the Ricci curvature of the Riemannian metrics induced by the evolving $G_2$-structures, $T$ is their torsion and $T^t$ its transpose. Using abstract index notation, $\mathsf V T_k=T_{ab}\varphi_{abk}$, and $\diamond \varphi$ acts on symmetric $2$-tensors as Then, if t

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.1: Corollary 5.18 in flows2
  • Definition 2.2
  • Proposition 2.3: Proposition 6.42 in flows2
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.1
  • ...and 42 more