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<p>We investigate generalisations of Hitchin’s functionals, whose critical points correspond to nearly Kähler and nearly parallel <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(G_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-structures. We focus on the gradient flow of these functionals and the spectral decomposition of their Hessians with respect to natural indefinite inner products. We introduce a Morse-like index for these functionals, termed the Hitchin index. We prove this index provides a lower bound for the Einstein co-index and explore its relationship with the deformation theory of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(G_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\operatorname {Spin}(7)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Spin</mo> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-conifolds.</p>

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Stability of Nearly Kähler and Nearly Parallel \(G_2\)-Manifolds

  • 期刊论文

  • 发表日期：

    2026年5月5日

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  • Enric Solé-Farré

摘要

We investigate generalisations of Hitchin’s functionals, whose critical points correspond to nearly Kähler and nearly parallel \(G_2\) G 2 -structures. We focus on the gradient flow of these functionals and the spectral decomposition of their Hessians with respect to natural indefinite inner products. We introduce a Morse-like index for these functionals, termed the Hitchin index. We prove this index provides a lower bound for the Einstein co-index and explore its relationship with the deformation theory of \(G_2\) G 2 and \(\operatorname {Spin}(7)\) Spin ( 7 ) -conifolds.