Uniqueness of asymptotically conical gradient shrinking solitons in G2-Laplacian flow

We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the two solitons are isometric. The proof extends Kotschwar–Wang’s argument for uniqueness of AC gradient shrinking Ricci solitons. We additionally show that the symmetries of the G2-structure of an AC shrinker end are inherited from its asymptotic cone; under a mild assumption on the fundamental group, the symmetries of the asymptotic cone extend to global symmetries.