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

TL;DR

This paper investigates the geometric properties of the space of closed G2-structures, demonstrating the concavity of Hitchin's volume functional and the decreasing nature of the G2 Laplacian flow, with new examples of hypersymplectic manifolds.

Contribution

It establishes the geodesic concavity of Hitchin's volume functional and constructs new hypersymplectic manifold examples within the space of closed G2-structures.

Findings

Hitchin's volume functional is geodesically concave under certain conditions.

The G2 Laplacian flow decreases the length in the space.

New examples of hypersymplectic manifolds are constructed.

Abstract

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.