Variation formulae for the volume of coassociative submanifolds

TL;DR

This paper derives new mathematical formulae for how the volume of coassociative submanifolds changes under variations, incorporating $G_2$ geometry, torsion, and Ricci curvature, with applications to fibrations.

Contribution

It introduces novel variation formulae for coassociative submanifolds' volume, including second variation within the moduli space, emphasizing ambient geometric data.

Findings

Derived first and second variation formulae in terms of $G_2$ data

Applied formulae to examples including fibrations

Highlighted the influence of torsion and Ricci curvature

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.