The Ricci Flow for Nilmanifolds

TL;DR

This paper studies the Ricci flow on simply connected nilmanifolds by analyzing the evolution of metric Lie algebra structures, providing systems of ODEs and explicit soliton solutions to understand their geometric behavior.

Contribution

It introduces a framework for analyzing Ricci flow on nilmanifolds via metric Lie algebra evolution and presents explicit solutions for soliton metrics.

Findings

Development of ODE systems describing Ricci flow on nilpotent Lie algebras

Explicit solutions for soliton metrics under Ricci flow

Qualitative analysis of the flow's properties on nilmanifolds

Abstract

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.