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A G₂-Hilbert functional in G₂-geometry

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arxiv 2505.06872 v1 pith:A6H2IQV6 submitted 2025-05-11 math.DG

A G₂-Hilbert functional in G₂-geometry

classification math.DG
keywords functionalgeometryhilbertstructuresuniquelyallowsanaloguesbasic
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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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