Laplacian Flow of Closed G$$_$$-Structures Inducing Nilsolitons

We study the existence of left invariant closed \(G_2\)-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these \(G_2\)-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on \({\mathbb {R}}^7\) in a similar way as in Lauret (Commun Anal Geom 19(5):831–854, 2011) we prove that the underlying metrics \(g(t)\) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as \(t\) goes to infinity.