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

TL;DR

This paper investigates the evolution of closed G$_2$-structures under the Laplacian flow on warped product manifolds, providing conditions for solutions and constructing new examples of expanding solitons.

Contribution

It reinterprets the Laplacian flow as evolution equations on the base manifold and offers a method to construct immortal solutions and new expanding solitons.

Findings

Derived conditions for solution existence in the coupled flow

Constructed explicit immortal solutions on warped products

Generated new examples of expanding Laplacian solitons

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.