Laplacian solitons and symmetry in G2-geometry

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a G2-structure, soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the space of symmetries (vector fields that annihilate via the Lie derivative) of a torsion-free G2-structure on a compact 7-manifold is canonically isomorphic to H1(M,R) by a direct method. Some comparisons with Ricci solitons are also discussed and emphasized, along with some future directions of exploration.