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

Abstract

We initiate a systematic study of cohomogeneity-one solitons in Bryant's Laplacian flow of closed G_2-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 Sp(2) and SU(3); 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 SU(3)-invariant steady Laplacian solitons defined on a neighbourhood of the zero-section of the anti-self-dual bundle of CP^2 is parametrised by the set of nonnegative reals. An open interval I=(0,c) corresponds to complete nontrivial gradient solitons that are asymptotic to the unique SU(3)-invariant torsion-free G_2 cone. The boundary point 0 of I corresponds to the well-known Bryant--Salamon asymptotically conical G_2-manifold, while the other boundary point c corresponds to an explicit complete gradient steady soliton with exponential volume growth and novel asymptotic geometry. The open interval (c, oo) consists entirely of incomplete solutions. In addition, we find an explicit complete gradient shrinking soliton on the anti-self-dual bundle of S^4 and CP^2. Both these shrinkers are asymptotic to closed but non-torsion-free G_2 cones. Like the nontrivial AC gradient steady solitons on the anti-self-dual bundle of CP^2, these shrinkers appear to be potential singularity models for finite-time singularities of Laplacian flow.