Laplacian flow of closed $G_2$-structures inducing nilsolitons

TL;DR

This paper investigates the Laplacian flow of closed $G_2$-structures on nilpotent Lie groups, proving long-term existence, uniqueness, and convergence of the flow to flat metrics, thus advancing understanding of geometric flows in special holonomy contexts.

Contribution

It establishes the long-time existence, uniqueness, and convergence of the Laplacian flow for closed $G_2$-structures inducing Ricci solitons on nilpotent Lie groups, linking geometric flow behavior to algebraic structures.

Findings

Flow exists uniquely for all time

Metrics converge to flat geometry

Flow behavior is characterized on Lie algebra level

Abstract

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 $\R^7$ in a similar way as in [23] 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.