A $G_2$-Hilbert functional in $G_2$-geometry

TL;DR

This paper introduces the $G_2$-Hilbert functional, a new variational tool in $G_2$-geometry, leading to novel flows analogous to Ricci flow and characterizing special $G_2$-structures.

Contribution

It defines a new $G_2$-Hilbert functional inspired by Einstein-Hilbert principles and introduces two new $G_2$-structure flows similar to Ricci flow.

Findings

Torsion-free and nearly $G_2$-structures are saddle points of the functional.

Two new $G_2$-structure flows are proposed, analogous to Ricci flow.

The functional distinguishes special $G_2$-structures through variational properties.

Abstract

In this paper we introduce a new functional on the space of $G_2$-structures which we call the $G_2$-Hilbert functional. It is uniquely determined by a few basic principles inspired by the Einstein-Hilbert functional in Riemannian Geometry, and it has similar variational behaviour with it. For instance, torsion-free and nearly $G_2$-structures are saddle critical points of the volume-normalized $G_2$-Hilbert functional. This allows us to uniquely distinguish two new flows of $G_2$-structures, which can be considered as analogues of the Ricci flow in $G_2$-geometry.