Variation formulae for the volume of coassociative submanifolds (Q6552599)

The paper under review deals with the calculus of variations of the volume functional for submanifolds, in a specific geometric context. More precisely, the purpose of this paper is to show how the above plays out in the context of \(G_2\) geometry: one deals with 7-dimensional manifolds endowed with a certain non-degenerate closed 3-form \(\phi\). The authors discuss the analogy between \(G_2\) and Kähler geometry, and point out the differences between them.\par The main result is a new second variation formula, specific to coassociative submanifolds. One standard goal of such a formula is to allow to detect geometric conditions ensuring that a minimal submanifold is also stable. The authors find sufficient conditions.\par The new formulae are illustrated with several examples, both homogeneous and non-homogeneoues.