Paper · May 11, 2025

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

Panagiotis Gianniotis, George Zacharopoulos

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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.

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

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A G2​-Hilbert functional in G2​-geometry

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