$S^1$-invariant Laplacian flow

Abstract

The Laplacian flow is a geometric flow introduced by Bryant as a way for finding torsion free $G_2$-structures. If the flow is $S^1$-invariant then it descends to a flow of $SU(3)$-structures on a $6$-manifold. In this article we derive expressions for these evolution equations. In our search for examples we discover the first inhomogeneous shrinking solitons, which are also gradient. We also show that any compact non-torsion free soliton admits no infinitesimal symmetry.

arXiv comment20 pages, to appear in The Journal of Geometric Analysis

References in corpus (10)

• The geometry of three-forms in six and seven dimensions
• The Ricci tensor of SU(3)-manifolds
• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• Kaehler reduction of metrics with holonomy G_2
• Laplacian flow of homogeneous G2-structures and its solitons
• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
• Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature
• Spacelike mean curvature flow
• On the automorphism group of a closed G${}_2$-structure
• A class of eternal solutions to the G${}_2$-Laplacian flow

Cited by in corpus (3)

• Spin(7) metrics from Kähler Geometry
• Special solutions to the Type IIA flow
• Closed G${}_2$-structures on unimodular Lie algebras with non-trivial center