The space of closed $G2$-structures. II. concavity and hypersymplectic structures

In the space of closed $G_2$-structures equipped with Bryant’s Dirichlet-type metric, we continue to utilise the geodesic, constructed in our previous article, to show that, under a normalisation condition Hitchin’s volume functional is geodesically concave and the $G_2$ Laplacian flow decreases the length. Furthermore, we also construct various examples of hyper-symplectic manifolds on geodesic concavity.