Solutions and singularities of the Ricci-harmonic flow and Ricci-like flows of $\mathrm{G_2}$-structures

TL;DR

This paper constructs explicit solutions and analyzes singularities of Ricci-harmonic and Ricci-like flows of G_2-structures on 7-dimensional manifolds, providing the first examples of certain singularity types.

Contribution

It introduces explicit solutions and classifies singularities of Ricci-harmonic and Ricci-like flows of G_2-structures, including the first examples of specific singularity types.

Findings

Found ancient solutions with Type I singularity on contact Calabi-Yau manifolds.

Identified immortal solutions with Type III or Type IIb singularities depending on curvature.

Extended solutions to flows on the 7-dimensional Heisenberg group.

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.