Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

TL;DR

This paper establishes the uniqueness of asymptotically conical gradient shrinking solitons in G_2-Laplacian flow, showing that such solitons asymptotic to the same cone are equivalent and inherit symmetries from their asymptotic cones.

Contribution

It extends the uniqueness results for AC gradient shrinking Ricci solitons to G_2-Laplacian flow and analyzes symmetry inheritance from asymptotic cones.

Findings

Uniqueness of AC gradient shrinking solitons in G_2-Laplacian flow.

Equivalence and isometry of solitons asymptotic to the same cone.

Symmetries of the G_2-structure are inherited from the asymptotic cone.

Abstract

We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G_2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G_2-cone, then their G_2-structures are equivalent, and in particular, the two solitons are isometric. The proof extends Kotschwar and Wang's argument for uniqueness of AC gradient shrinking Ricci solitons. We additionally show that the symmetries of the G_2-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.