Cohomogeneity-one solitons in Laplacian flow: Local, smoothly-closing and steady solitons

We initiate a systematic study of cohomogeneity-one solitons in Bryant's Laplacian flow of closed (Formula presented.) -structures on a 7-manifold, motivated by the problem of understanding finite-time singularities of that flow. Here, we focus on solitons with symmetry groups (Formula presented.) and (Formula presented.); in both cases, we prove the existence of continuous families of local cohomogeneity-one gradient Laplacian solitons and characterise which of these local solutions extend smoothly over their unique singular orbits. The main questions are then to determine which of these smoothly-closing solutions extend to complete solitons and furthermore to understand the asymptotic geometry of these complete solitons. We provide complete answers to both questions in the case of steady solitons. Up to the actions of scaling and discrete symmetries, we show that the set of all smoothly-closing (Formula presented.) -invariant steady Laplacian solitons defined on a neighbourhood of the zero section of (Formula presented.) is parametrised by (Formula presented.), the set of non-negative reals. We then determine precisely which of these solutions extend to a complete soliton defined on the whole of (Formula presented.). An open interval (Formula presented.) corresponds to complete non-trivial gradient solitons that are asymptotic to the unique (Formula presented.) -invariant torsion-free (Formula presented.) -cone. The point (Formula presented.) corresponds to the well-known Bryant–Salamon asymptotically conical torsion-free structure on (Formula presented.) viewed as a trivial steady soliton, while the other point (Formula presented.) corresponds to an explicit complete gradient steady soliton with exponential volume growth and novel asymptotic geometry. The open interval (Formula presented.) consists entirely of incomplete solutions. In addition, we find an explicit complete gradient shrinking soliton on (Formula presented.) and (Formula presented.). Both these shrinkers are asymptotic to closed but non-torsion-free (Formula presented.) -cones. Like the non-trivial AC gradient steady solitons on (Formula presented.), these shrinkers appear to be potential singularity models for finite-time singularities of Laplacian flow. We also compare the behaviour of the Laplacian solitons we construct to solitons in Ricci flow.