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Solutions of the Laplacian flow and coflow of a locally conformal parallel \(\mathrm {G}_2\)-structure

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Abstract

We study the Laplacian flow of a \(\mathrm {G}_2\)-structure where this latter structure is claimed to be locally conformal parallel. The first examples of long time solutions of this flow with the locally conformal parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. These examples are one-parameter families of locally conformal parallel \(\mathrm {G}_2\)-structures on rank-one solvable extensions of six-dimensional nilpotent Lie groups. The found solutions are used to construct long time solutions to the Laplacian coflow starting from a locally conformal parallel structure. We also study the behavior of the curvature of the solutions obtaining that for one of the examples the induced metric is Einstein along all the flow (resp. coflow).

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Acknowledgements

The authors would like to thank Anna Fino and Luis Ugarte for useful comments on the subject. This work has been partially supported by the Projects MTM2017-85649-P (AEI/FEDER, UE), E22-17R “Álgebra y Geometría” (Gobierno de Aragón/FEDER), and UZCUD2019-CIE-02 “Nuevos ejemplos de variedades en dimensiones 6 y 7 con geometrías especiales” (Centro Universitario de la Defensa de Zaragoza, Academia General Militar). The third author would also like to thank the Fields Institute for its support during her stay in Toronto.

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Authors and Affiliations

  1. Departamento de Matemáticas - I.U.M.A., Facultad de Ciencias Humanas y de la Educación, Universidad de Zaragoza, 22003, Huesca, Spain

    Victor Manero

  2. Centro Universitario de la Defensa - I.U.M.A., Academia General Militar, Crta. de Huesca s/n., 50090, Zaragoza, Spain

    Antonio Otal & Raquel Villacampa

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  1. Victor Manero
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  2. Antonio Otal
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Correspondence to Victor Manero.

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Appendix

Appendix

See Tables 4 and 5.

Table 4 Non-vanishing coefficients of the curvature of the metric \(g_t\) induced by the solutions of the LCP flow expresssed in the adapted basis \(\{x_i\}_{i=1}^7\)
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Table 5 Continuation of Table 4
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Manero, V., Otal, A. & Villacampa, R. Solutions of the Laplacian flow and coflow of a locally conformal parallel \(\mathrm {G}_2\)-structure. manuscripta math. 165, 61–87 (2021). https://doi.org/10.1007/s00229-020-01205-2

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