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Variation formulae for the volume of coassociative submanifolds

Tommaso Pacini, Alberto Raffero

TL;DR

This work develops new second-variation formulas for the volume of coassociative submanifolds in 7‑manifolds with a $G_2$ structure, expressing the variation in terms of ambient torsion and Ricci curvature. It situates the coassociative setting alongside the Lagrangian/Kähler picture, deriving a moduli-space–adapted second variation and highlighting how torsion obstructs stability unless controlled by Bryant’s quadratic condition. The paper then applies the theory to concrete contexts, including Extremally Ricci-Pinched (ERP) $G_2$ structures and coassociative fibrations, providing explicit computations and stability conclusions. It also develops perturbation schemes to generate non-homogeneous examples, illustrating how ambient torsion and curvature influence local minimality and the Hessian of fiber volumes. Overall, the results clarify stability criteria for coassociatives under various geometric regimes and enhance the understanding of calibrated-volume variations in $G_2$ geometry.

Abstract

We prove new variation formulae for the volume of coassociative submanifolds, expressed in terms of $G_2$ data. As a special case, we obtain a second variation formula for variations within the moduli space of coassociative submanifolds; this formula highlights the role of the ambient torsion and Ricci curvature. These results apply, for example, to coassociative fibrations. We illustrate our formulae with several examples, both homogeneous and non.

Variation formulae for the volume of coassociative submanifolds

TL;DR

This work develops new second-variation formulas for the volume of coassociative submanifolds in 7‑manifolds with a structure, expressing the variation in terms of ambient torsion and Ricci curvature. It situates the coassociative setting alongside the Lagrangian/Kähler picture, deriving a moduli-space–adapted second variation and highlighting how torsion obstructs stability unless controlled by Bryant’s quadratic condition. The paper then applies the theory to concrete contexts, including Extremally Ricci-Pinched (ERP) structures and coassociative fibrations, providing explicit computations and stability conclusions. It also develops perturbation schemes to generate non-homogeneous examples, illustrating how ambient torsion and curvature influence local minimality and the Hessian of fiber volumes. Overall, the results clarify stability criteria for coassociatives under various geometric regimes and enhance the understanding of calibrated-volume variations in geometry.

Abstract

We prove new variation formulae for the volume of coassociative submanifolds, expressed in terms of data. As a special case, we obtain a second variation formula for variations within the moduli space of coassociative submanifolds; this formula highlights the role of the ambient torsion and Ricci curvature. These results apply, for example, to coassociative fibrations. We illustrate our formulae with several examples, both homogeneous and non.
Paper Structure (18 sections, 12 theorems, 98 equations)

This paper contains 18 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Variation formulae in calibrated geometry.
  3. $\mathbf{G_2}$ vs. Kähler geometry.
  4. First variation formulae
  5. The classical setting.
  6. $\mathbf{G_2}$ manifolds and coassociative submanifolds.
  7. The first variation formula for coassociatives.
  8. Moduli spaces.
  9. Fibrations.
  10. Second variation formulae
  11. The classical setting.
  12. The second variation formula for coassociatives.
  13. The second variation formula for Lagrangians.
  14. Conclusions.
  15. Controlling the torsion
  16. ...and 3 more sections

Key Result

Lemma 2.1

$\frac{d}{dt}f(t)_{|t=0}=\hbox{div}_{\Sigma}(Z^\top)-H\cdot Z^\perp$.

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Remark
  • Proposition 3.1
  • Corollary 3.2
  • proof : Proof of Proposition $\ref{['2Classic']}$
  • ...and 27 more