Laplacian flow of homogeneous G2-structures and its solitons

Abstract

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.

33 pages, final version accepted in Proc. London Math. Soc

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• Stability of torsion-free G_2 structures along the Laplacian flow
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• A new example of a compact ERP G2-structure
• $S^1$-invariant Laplacian flow
• Remarks on homogeneous solitons of the G${}_2$-Laplacian flow
• Harmonic $G_2$-structures on almost Abelian Lie groups
• On $G_2$-structures, special metrics and related flows
• A report on the hypersymplectic flow
• 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