A class of eternal solutions to the G$_2$-Laplacian flow
Anna Fino, Alberto Raffero
TL;DR
The paper addresses the G$_2$-Laplacian flow starting from an extremally Ricci-pinched (ERP) closed ${\rm G}_2$-structure on a seven-manifold. It reduces the flow to an ODE for an auxiliary function when the intrinsic torsion $\tau$ has constant norm, obtaining the explicit eternal solution $\varphi(t)=\varphi+f(t)\,d\tau$ with $f(t)=\tfrac{6}{|\tau|_\varphi^2}\big(e^{|\tau|_\varphi^2 t/6}-1\big)$ and $\tau(t)=e^{|\tau|_\varphi^2 t/6}\tau$, which ensures long-time existence and preservation of the ERP condition. The authors show the Ricci tensor stays constant along the flow and describe the metric evolution by $\partial_t g_{\varphi(t)}=\frac{|\tau|_\varphi^2}{6}\,g_{\varphi(t)}|_Q$, with volume scaling $dV_{\varphi(t)}=e^{|\tau|_\varphi^2 t/3} dV_{\varphi}$. In the compact case, they analyze asymptotics of volumes and leaves, showing $P$-leaves collapse while $Q$-leaves and the manifold volume grow, and they discuss concrete ERP examples (Bryant and Lauret) and a unimodular uniqueness result. Overall, the work provides explicit, globally defined ERP-preserving solutions to the Laplacian flow and clarifies long-time behavior and geometric structure under ERP hypotheses. The results contribute to understanding how special G$_2$-geometries evolve under natural flows and offer templates for constructing metrics with prescribed torsion and curvature properties.
Abstract
We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. This result holds more generally on any 7-manifold whenever the intrinsic torsion of the extremally Ricci-pinched G$_2$-structure has constant norm. We also discuss various examples.