<p>In this paper we introduce a new functional on the space of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-structures which we call the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>–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 to it. For instance, torsion-free and nearly <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-structures are saddle critical points of the volume-normalized <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>–Hilbert functional. This allows us to uniquely distinguish two new flows of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-structures, which can be considered as analogues of the Ricci flow in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$G_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-geometry.</p>

A $G_2$-Hilbert Functional in $G_2$-Geometry

期刊论文
发表日期：

2026年1月4日

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Panagiotis Gianniotis,
George Zacharopoulos

摘要

In this paper we introduce a new functional on the space of $G_2$ $G_{2}$ -structures which we call the $G_2$ $G_{2}$ –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 to it. For instance, torsion-free and nearly $G_2$ $G_{2}$ -structures are saddle critical points of the volume-normalized $G_2$ $G_{2}$ –Hilbert functional. This allows us to uniquely distinguish two new flows of $G_2$ $G_{2}$ -structures, which can be considered as analogues of the Ricci flow in $G_2$ $G_{2}$ -geometry.

A \(G_2\)-Hilbert Functional in \(G_2\)-Geometry

摘要