Laplacian Solitons and Symmetry in G₂-geometry

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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). Some comparisons with Ricci solitons are also discussed, along with some future directions of exploration.