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The \(\textrm{G}_2\) Geometry of 3-Sasaki Structures

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Abstract

We initiate a systematic study of the deformation theory of the second Einstein metric \(g_{\frac{1}{\sqrt{5}}}\) respectively the proper nearly \(\textrm{G}_2\) structure \(\varphi _{\frac{1}{\sqrt{5}}}\) of a 3-Sasaki manifold \((M^7,g)\). We show that infinitesimal Einstein deformations for \(g_{\frac{1}{\sqrt{5}}}\) coincide with infinitesimal \(\textrm{G}_2\) deformations for \(\varphi _{\frac{1}{\sqrt{5}}}\). The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of g, with eigenvalue twice the Einstein constant of the 4-dimensional base orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal \(\textrm{G}_2\) deformations which are unobstructed to second order.

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Acknowledgements

Paul-Andi Nagy was supported by the Institute for Basic Science (IBS-R032-D1).This research has also been supported by the Special Priority Program SPP 2026 ‘Geometry at Infinity’ funded by the DFG. It is a pleasure to thank Tommy Murphy for many useful conversations on stability.

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Authors and Affiliations

  1. Center for Complex Geometry, Institute for Basic Science(IBS), 55 Expo-ro, Daejeon, Yuseong-gu, 34126, South Korea

    Paul-Andi Nagy

  2. Institut für Geometrie und Topologie Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Uwe Semmelmann

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  1. Paul-Andi Nagy
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Nagy, PA., Semmelmann, U. The \(\textrm{G}_2\) Geometry of 3-Sasaki Structures. J Geom Anal 34, 61 (2024). https://doi.org/10.1007/s12220-023-01494-5

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