Ricci flow on three-dimensional, unimodular metric Lie algebras

David A Glickenstein, Tracy L. Payne

Research output: Contribution to journalArticle

19 Scopus citations

Abstract

We give a global picture of the Ricci flow on the space of threedimensional, 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-bycase 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.

Original languageEnglish (US)
Pages (from-to)927-961
Number of pages35
JournalCommunications in Analysis and Geometry
Volume18
Issue number5
Publication statusPublished - Dec 2010

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ASJC Scopus subject areas

  • Statistics and Probability
  • Geometry and Topology
  • Analysis
  • Statistics, Probability and Uncertainty

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