Flows of $\mathrm{G}_2$-structures on contact Calabi--Yau $7$-manifolds

Abstract

We study the Laplacian flow and coflow on contact Calabi-Yau $7$-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though not eternal and not self-similar) solution of the Laplacian coflow which has an infinite time singularity, that is Type IIb unless the transverse Calabi--Yau geometry is flat. The flows in each case collapse (after normalising the volume) to a lower-dimensional limit, which is either $\mathbb{R}$ for the Laplacian flow or standard $\mathbb{C}^3$ for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow up to reparametrisation of time, and defines an (incomplete) Calabi--Yau structure on the spacetime track of the flow.