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

Panagiotis Gianniotis; George Zacharopoulos

paper

A $G_{2}$ -Hilbert functional in $G_{2}$ -geometry

arXiv:2505.06872

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.

Comments are welcome

A G2​-Hilbert functional in G2​-geometry

Abstract

A $G_{2}$ -Hilbert functional in $G_{2}$ -geometry