Abstract
The space of \(G_2\)-structures is naturally stratified by those structures compatible with a fixed Riemannian metric. We study the restriction of the total torsion energy functional to these strata. Precisely we show the short-time existence of its negative gradient flow, we characterise the space of its critical points in terms of spinor fields and, finally, we describe the long-time behaviour of the homogeneous negative gradient flow.
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Notes
As already explained at the end of the introduction, we identify vector fields with their Riesz duals.
Here the starred operators denote the formal adjoint ones.
See the definition of \(R^{\omega }\).
Note that any G-invariant \(D^2\)-eigenvector has constant length.
References
Ammann, B., Weiss, H., Witt, F.: A spinorial energy functional: critical points and gradient flow. Math. Ann. 365(3), 1559–1602 (2016)
Agricola, I., Chiossi, G.S., Friedrich, T., Höll, J.: Spinorial description of \(\text{ SU }(3)\) and \(\text{ G }_2\)-structures. J. Geom. Phys. 98, 535–555 (2015)
Bedulli, L., Vezzoni, L.: A parabolic flow of balanced metrics. J. Reine Angew. Math. 723, 79–99 (2017)
Bryant, R.L.: Some remarks on \(\text{ G }_2\)-structures. In: Proceedings of Gökova Geometry-Topology Conference (2005)
Bryant, R.L., Xu, F.: Laplacian flow for closed \(\text{ G }_2\)-structures: short time behavior. arXiv:1101.2004 (2011)
Friederich, T.: Dirac Operators in Riemannian Geometry Graduate Studies in Mathematics, vol. 25. AMS, Providence, RI (2000)
Grigorian, S.: Short-time behaviour of a modified Laplacian coflow of \(\rm G_2-\)structures. Adv. Math. 248, 378–415 (2013)
Grigorian, S.: \(G_2\)-structures and octonion bundles. Adv. Math. 308, 142–207 (2017)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc 7(1), 65–222 (1982)
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)
Hitchin, N.: The geometry of three-forms in six and seven dimensions. arXiv:math/0010054 (2000)
Lawson, H.B., Michelsohn, M.L.: Spin Geometry. Princeton University Press, Princeton (1989)
Weiss, H., Witt, F.: A heat flow for special metrics. Adv. Math. 231(6), 3288–3322 (2012)
Acknowledgements
The author is grateful to Luigi Vezzoni and the anonymous referees for many useful comments and improvements. Also, he wants to thank Alberto Raffero and Francesco Pediconi, who shared so many pleasant conversations with him.
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Bagaglini, L. The Energy Functional of \(G_2\)-Structures Compatible with a Background Metric. J Geom Anal 31, 346–365 (2021). https://doi.org/10.1007/s12220-019-00264-6
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DOI: https://doi.org/10.1007/s12220-019-00264-6