Laplacian Flow of Closed G $$_2$$ -Structures Inducing Nilsolitons
Tóm tắt
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.
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