Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry

Abstract

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt}\psi =-{\Delta }_d\psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\varphi$ and ${\Delta }_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($d\psi =0$). We study this flow for two explicit examples of coclosed $G_2$-structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold $N$ which is taken to be either a nearly Kähler manifold or a Calabi-Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi-Yau case, we find all the soliton solutions explicitly. In the nearly Kähler case, we find several special soliton solutions, and reduce the general problem to a single \emph{third order} highly nonlinear ordinary differential equation.

18 pages, no figures. Version 2: Minor improvements and addition of references. To appear in Differential Geometry and its Applications

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• Laplacian flow of homogeneous G2-structures and its solitons
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• Laplacian Solitons and Symmetry in G_2-geometry
• A parabolic flow of balanced metrics
• A gradient flow of isometric ${\mathrm{G}}_2$ structures
• Estimates and monotonicity for a heat flow of isometric G2-structures
• Modified Laplacian coflow of $G_2$-structures on manifolds with symmetry
• Flows of ${\mathrm{G}}_2$-structures on contact Calabi--Yau $7$-manifolds
• A new example of a compact ERP G2-structure
• Coupled SU(3)-structures and Supersymmetry
• Geometric Flows of $G_2$-Structures on 3-Sasakian 7-Manifolds
• Explicit Soliton for the Laplacian Co-Flow on a Solvmanifold
• Harmonic $G_2$-structures on almost Abelian Lie groups
• On $G_2$-structures, special metrics and related flows
• Some remarks on strong ${\mathrm{G}}_2$-structures with torsion
• Local curvature estimates for the Laplacian flow
• Moduli spaces of $G_2$ and $Spin(7)-$instantons on product manifolds
• Cohomogeneity one solitons for the isometric flow of $G_2$-structures
• Real analyticity of the modified Laplacian coflow