Laplacian flow for closed \(\mathrm{G}_2\) structures: real analyticity (Q2633484)

Let \(M\) be a compact \(7\)-manifold and let \(\varphi _{0}\) be a closed \(G_2\) structure on \(M\). In the paper under review the authors consider solutions \(\varphi (t)\), \(t\in \lbrack 0,T_0]\) to the Laplacian flow for closed \(G_2\) structures: \[ \begin{cases} \frac{\partial }{\partial t}\varphi =\Delta _{\varphi }\varphi , \\ d\varphi =0, \\ \varphi (0)=\varphi _{0}, \end{cases} \] where \(\Delta _{\varphi }\varphi =dd^{\ast }\varphi +d^{\ast }d\varphi \) is the Hodge Laplacian of \(\varphi (t)\) with respect to the metric \(g(t)\) determined by \(\varphi (t)\). The authors show that for each fixed time \(t\in (0,T_{0}]\), \((M,\varphi (t),g(t))\) is real analytic, where \(g(t)\) is the metric induced by \(\varphi (t)\). Consequently, any Laplacian soliton is real analytic and they obtain unique continuation results for the flow.