Laplacian Solitons and Symmetry in G_2-geometry

Abstract

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-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 $G_2$-structure on a compact 7-manifold is canonically isomorphic to $H^1(M,\mathbb{R})$. Some comparisons with Ricci solitons are also discussed, along with some future directions of exploration.

References in corpus (3)

• Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry
• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
• Energy functionals and soliton equations for G_2-forms

Cited by in corpus (10)

• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• Laplacian flow of homogeneous G2-structures and its solitons
• A class of eternal solutions to the G${}_2$-Laplacian flow
• Remarks on homogeneous solitons of the G${}_2$-Laplacian flow
• Laplacian solitons: questions and homogeneous examples
• Curvature pinching estimate under the Laplacian G_{2} flow
• Locally conformal calibrated $G_2$-manifolds
• Torsion-free $G_2$-structures with identical Riemannian metric
• Closed G${}_2$-structures on unimodular Lie algebras with non-trivial center
• On the existence of homogeneous solitons of gradient type for the G${}_2$-Laplacian flow