Convergence and stability of locally \mathbb{R}^{N}-invariant solutions of Ricci flow

TL;DR

This paper studies the convergence and stability of locally -invariant Ricci flow solutions on principal bundles, providing insights into their long-term behavior and implications for 3D Ricci flow classification.

Contribution

It establishes convergence and asymptotic stability of R^{N}-invariant solutions, extending understanding of Ricci flow dynamics on principal bundles with nilpotent symmetry.

Findings

Proves convergence of certain R^{N}-invariant solutions

Shows stability modulo finite-dimensional center manifolds

Connects results to classification of 3D Ricci flow solutions

Abstract

Important models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G-invariant solutions on principal bundles, where G is a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain R^{N}-invariant solutions. When the dimension of the total space is three, these results are relevant to work of Lott classifying the asymptotic behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures and diameters are respectively O(t^{-1}) and O(t^{1/2}).