Abstract
The Laplacian flow is a geometric flow introduced by Bryant as a way for finding torsion free \(G_2\)-structures starting from a closed one. If the flow is invariant under a free \(S^1\) action then it descends to a flow of SU(3)-structures on a 6-manifold. In this article we derive expressions for these evolution equations. In our search for examples we discover the first inhomogeneous shrinking solitons, which are also gradient. We also show that any compact non-torsion free soliton admits no infinitesimal symmetry.
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Notes
Here we are using Salamon’s notation [19] to mean that the Lie algebra admits a coframing \(e^i\) with \(de^i=e^{jk}\), where jk denotes the ith entry.
References
Lotay, J.D., Wei, Y.: Laplacian flow for closed \(G_2\) structures: shi-type estimates, uniqueness and compactness. Geom. Funct. Anal. 27(1), 165–233 (2017)
Bryant, R.L.: Some remarks on \(G_2\)-structures. In: Proceedings of Gökova Geometry-Topology Conference 2005. International Press (2006)
Bryant, R.L., Xu, F.: Laplacian flow for closed \(G_2\)-structures: short time behavior. ar**v:1101.2004 (2011)
Fine, J., Yao, C.: Hypersymplectic 4-manifolds, the \({G_2}\)-Laplacian flow, and extension assuming bounded scalar curvature. Duke Math. J. 167(18), 3533–3589 (2018)
Fino, A., Raffero, A.: Closed warped \(G_2\)-structures evolving under the Laplacian flow. Ann. Sc. Norm. Super. 20, 315–348 (2017)
Fino, A., Raffero, A.: A class of eternal solutions to the \(G_2\) Laplacian flow. J. Geom. Anal. 31, 4641–4660 (2020)
Lambert, B., Lotay, J.D.: Spacelike mean curvature flow. J. Geom. Anal. 31, 1291–1359 (2019)
Lauret, J.: Laplacian flow of homogeneous \(G_2\)-structures and its solitons. Proc. Lond. Math. Soc. 114(3), 527–560 (2017)
Nicolini, M.: New examples of shrinking Laplacian solitons. Q. J. Math. (2021). https://doi.org/10.1093/qmath/haab029
Apostolov, V., Salamon, S.: Kähler reduction of metrics with holonomy \({G_2}\). Commun. Math. Phys. 246(1), 43–61 (2004)
Hitchin, N.: The geometry of three-forms in six and seven dimensions. J. Differ. Geom. 55(3), 547–576 (2000)
Chiossi, S., Salamon, S.: The intrinsic torsion of \(SU(3)\) and \(G_2\) structures. In: Differential Geometry. Valencia 2001, pp. 115–133. World Scientific, Singapore (2002)
Salamon, S.: Riemannian Geometry and Holonomy Groups. Longman Scientific and Technical, Harlow (1989)
Bedulli, L., Vezzoni, L.: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57(4), 1125–1146 (2007)
Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126(3), 525–576 (1987)
Kobayashi, S.: Principal fibre bundles with the \(1\)-dimensional toroidal group. Tohoku Math. J. 8(1), 29–45 (1956)
Joyce, D.: Riemannian Holonomy Groups and Calibrated Geometry, vol. 12. Oxford University Press, Oxford (2007)
Podestà, F., Raffero, A.: On the automorphism group of a closed \(G_2\)-structure. Q. J. Math. 70(1), 195–200 (2018)
Salamon, S.: Complex structures on nilpotent Lie algebras. J. Pure Appl. Algebra 157(2), 311–333 (2001)
Fernández, M.: An example of a compact calibrated manifold associated with the exceptional Lie group \(G_2\). J. Differ. Geom. 26(2), 367–370 (1987)
Fernández, M., Fino, A., Manero, V.: Laplacian flow of closed \(G_2\)-structures inducing nilsolitons. J. Geom. Anal. 26(3), 1808–1837 (2016)
Ball, G.: Quadratic closed \(G_2\)-structures. ar**v:2006.14155 (2020)
Acknowledgements
The author is indebted to his PhD advisors Jason Lotay and Simon Salamon for their constant support and many helpful discussions that led to this article. The author would also like to thank Andrew Dancer and Lorenzo Foscolo for helpful comments on a version of this paper that figures in the author’s thesis. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.
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Fowdar, U. \(S^1\)-Invariant Laplacian Flow. J Geom Anal 32, 17 (2022). https://doi.org/10.1007/s12220-021-00784-0
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DOI: https://doi.org/10.1007/s12220-021-00784-0