Ricci flow on three-dimensional, unimodular metric Lie algebras

TL;DR

This paper provides a comprehensive phase plane analysis of Ricci flow on three-dimensional unimodular metric Lie algebras, unifying previous case-by-case studies and identifying special algebraic structures like Ricci solitons.

Contribution

It introduces a global dynamical systems approach to Ricci flow on these algebras, characterizing fixed points and special trajectories systematically.

Findings

Fixed points correspond to special metric Lie algebras.

Special trajectories include Ricci solitons and Riemannian submersions.

Unified framework simplifies previous case-by-case analyses.

Abstract

We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving orthonormal frame. This system is amenable to direct phase plane analysis, and we find that the fixed points and special trajectories in the phase plane correspond to special metric Lie algebras, including Ricci solitons and special Riemannian submersions. These results are one way to unify the study of Ricci flow on left invariant metrics on three-dimensional, simply-connected, unimodular Lie groups, which had previously been studied by a case-by-case analysis of the different Bianchi classes. In an appendix, we prove a characterization of the space of three-dimensional, unimodular, nonabelian metric Lie algebras modulo isometry and scaling.