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

Shubham Dwivedi, Ragini Singhal

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.

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 -structures, the Ricci-like flows of -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 -structures, on -dimensional contact Calabi-Yau manifolds and the -dimensional Heisenberg group. We prove that the natural co-closed -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 -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 -dimensional Heisenberg group.
Paper Structure (11 sections, 17 theorems, 136 equations)

This paper contains 11 sections, 17 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. $\mathrm{G}_2$-geometry and contact Calabi-Yau manifolds
  3. $\mathrm{G}_2$-geometry
  4. Contact Calabi-Yau manifolds
  5. Ansatz for the flows
  6. Solutions to flows of $\mathrm{G}_2$-structures on contact Calabi-Yau manifolds
  7. Ricci-harmonic flow
  8. Ricci-like flows in panos-georgeG2Hilbert
  9. Negative gradient flow of an energy functional
  10. Solutions of the flows on the 7-dimensional Heisenberg group
  11. Ansatz for the flows

Key Result

Theorem 1.1

Let $\varphi(t)$ be a solution of the Ricci-harmonic flow on a closed $7$-manifold $M$ on a maximal time interval $[0, \tau)$. We define the following quantity along the flow. Then Moreover, the quantity $\Lambda(t)$ blows-up at the following rate, where $C>0$ is a constant.

Theorems & Definitions (37)

  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 27 more