Laplacian Flow of Closed G$$_2$$2-Structures Inducing Nilsolitons

By M. Fernández, Anna Fino, V. Manero. 2013

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We study the existence of left invariant closed $$G_2$$G2-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $$G_2$$G2-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 $${\mathbb {R}}^7$$R7 in a similar way as in Lauret (Commun Anal Geom 19(5):831–854, 2011) we prove that the underlying metrics $$g(t)$$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 $$t$$t goes to infinity.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e7dd7581bc368b3058fd66542a82567a028b55d5