The generality of closed $\mathrm{G}_2$ solitons

The local generality of the space of solitons for the Laplacian flow of closed $\mathrm{G}_2$-structures is analyzed, and it is shown that the germs of such structures depend, up to diffeomorphism, on $16$ functions of $6$ variables (in the sense of É. Cartan). The method is to construct a natural exterior differential system whose integral manifolds describe such solitons and to show that it is involutive in Cartan’s sense, so that Cartan–Kähler theory can be applied.

Meanwhile, it turns out that, for the more special case of gradient solitons, the natural exterior differential system is not involutive, and the generality of these structures remains a mystery.

Keywords

$G_2$-structures, solitons

2010 Mathematics Subject Classification

Primary 53-xx. Secondary 53C29.

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The author thanks the Simons Foundation for its support via the Simons Collaboration Grant “Special Holonomy in Geometry, Analysis, and Physics.”

Received 30 April 2022

Accepted 12 August 2022

Published 30 January 2024