Explicit soliton for the Laplacian co-flow on a solvmanifold

  • Andrés J. MorenoEmail author
  • Henrique N. Sá Earp
Special Section: An Homage to Manfredo P. do Carmo


We apply the general Ansatz proposed by Lauret (Rend Semin Mat Torino 74:55–93, 2016) for the Laplacian co-flow of invariant \(\mathrm {G}_2\)-structures on a Lie group, finding an explicit soliton on a particular almost Abelian 7–manifold. Our methods and the example itself are different from those presented by Bagaglini and Fino (Ann Mat Pura Appl 197(6):1855–1873, 2018).



  1. 1.
    Arroyo, R.: The Ricci flow in a class of solvmanifolds. Differ. Geom. Appl. 31(4), 472–485 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bagaglini, L., Fernández, M., Fino, A.: Laplacian coflow on the 7-dimensional Heisenberg group (2017). arXiv:1704.00295
  3. 3.
    Bagaglini, L., Fino, A.: The Laplacian coflow on almost-abelian Lie groups. Ann. Mat. Pura Appl. 197(6), 1855–1873 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bryant, R., Xu, F.: Laplacian flow for closed \({\rm G}_2 \)-structures: short time behavior (2011). arXiv:1101.2004
  5. 5.
    Bryant, R.: Some remarks on \({\rm G}_2 \)-structures. In: Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, pp. 75–109 (2006)Google Scholar
  6. 6.
    Corti, A., Haskins, M., Nordström, J., Pacini, T.: \({\rm G}_2 \)-manifolds and associative submanifolds via semi-Fano \(3\)-folds. Duke Math. J. 164(10), 1971–2092 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fernández, M., Fino, A., Manero, V.: \({\rm G}_2 \)-structures on Einstein solvmanifolds. Asian J. Math. 19, 321–342 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fernández, M., Gray, A.: Riemannian manifolds with structure group \({\rm G}_2 \). Ann. Mat. Pura Appl. (4) 132, 19–45 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grigorian, S.: Short-time behaviour of a modified Laplacian coflow of \({\rm G}_2 \)-structures. Adv. Math. 248, 378–415 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hitchin, N.: The geometry of three-forms in six and seven dimensions, pp. 1–38 (2008). arXiv:math/0010054
  11. 11.
    Joyce, D.: Compact Riemannian 7-manifolds with holonomy \(\rm G_2 \) I. J. Differ. Geom. 43, 291–328 (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Joyce, D., Karigiannis, S.: A new construction of compact \(\rm G_2 \)-manifolds by gluing families of Eguchi–Hanson spaces, J. Differ. Geom. (2017). arXiv:1707.09325 (to appear)
  13. 13.
    Karigiannis, S.: Flows of \({\rm G}_2 \)-structures. I. Q. J. Math. 60(4), 487–522 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Karigiannis, S., McKay, B., Tsui, M.: Soliton solutions for the Laplacian co-flow of some \({\rm G}_2\)-structures with symmetry. Differ. Geom. Appl. 30(4), 318–333 (2012)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lauret, J.: Geometric flows and their solitons on homogeneous spaces. Rend Semin. Mat. Torino 74, 55–93 (2016)MathSciNetGoogle Scholar

Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.University of Campinas (Unicamp)CampinasBrazil

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