Closed warped G${}_2$-structures evolving under the Laplacian flow

Abstract

We study the behaviour of the Laplacian flow evolving closed G${}_2$-structures on warped products of the form $M^6\times {\mathbb{S}}^1$, where the base $M^6$ is a compact 6-manifold endowed with an SU(3)-structure. In the general case, we reinterpret the flow as a set of evolution equations on $M^6$ for the differential forms defining the SU(3)-structure and the warping function. When the latter is constant, we find sufficient conditions for the existence of solutions of the corresponding coupled flow. This provides a method to construct immortal solutions of the Laplacian flow on the product manifolds $M^6\times {\mathbb{S}}^1$. The application of our results to explicit cases allows us to obtain new examples of expanding Laplacian solitons.

27 pages. To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

Cited by in corpus (3)

• Closed G${}_2$-structures on non-solvable Lie groups
• Homogeneous symplectic half-flat 6-manifolds
• On $G_2$-structures, special metrics and related flows