Solutions and singularities of the Ricci-harmonic flow and Ricci-like flows of ${\mathrm{G}}_2$-structures

Abstract

We find explicit solutions and singularities of the Ricci-harmonic flow of ${\mathrm{G}}_2$-structures, the Ricci-like flows of ${\mathrm{G}}_2$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of ${\mathrm{G}}_2$-structures, on $7$-dimensional contact Calabi-Yau manifolds and the $7$-dimensional Heisenberg group. We prove that the natural co-closed ${\mathrm{G}}_2$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity, and it gives an immortal solution to the Ricci-like flows with an infinite time singularity which are Type III if the transversal Calabi-Yau distribution is flat, and Type IIb otherwise. The same ansatz gives ancient solution to the negative gradient flow of ${\mathrm{G}}_2$-structures. These are the first examples of Type I singularities of the Ricci-harmonic flow and Type IIb and Type III singularities of the Ricci-like flows. We also obtain similar solutions for all the three flows on the $7$-dimensional Heisenberg group.

20 pages. All comments welcome