Solutions of the Laplacian flow and coflow of a locally conformal parallel $$\mathrm {G}_2$$-structure

Abstract

We study the Laplacian flow of a G 2 -structure where this latter structure is claimed to be Locally Conformal Parallel. The first examples of long time solutions of this flow with the Locally Conformal Parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. These examples are one-parameter families of Locally Conformal Parallel G 2 -structures on rankone solvable extensions of six-dimensional nilpotent Lie groups. The found solutions are used to construct long time solutions to the Laplacian coflow starting from a Locally Conformal Parallel structure. We also study the behavior of the curvature of the solutions obtaining that for one of the examples the induced metric is Einstein along all the flow (resp. coflow).

Flows of co-closed $G_{2}$-structures

“…Similarly, in [2], Bagaglini and Fino studied the Laplacian coflow on 7-dimensional almost-abelian Lie groups and showed long-term existence properties and constructed soliton solutions. In [29], Manero, Otal, and Villacampa studied both the Laplacian flow (4.1) and the coflow (1.4) on solvmanifolds, but instead of restricting to closed or co-closed G 2 -structures, they instead restricted to locally conformally parallel G 2 -structures, which are the ones where only the 7-dimensional τ 1 component of the torsion may be nonvanishing.…”