Convergence and Stability of Locally ℝ N -Invariant Solutions of Ricci Flow

Abstract

Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally \(\mathcal{G}\) -invariant solutions on bundles \(\mathcal{G}^{N}\hookrightarrow\mathcal{M}\,\overset{\pi }{\mathcal{\longrightarrow}}\,\mathcal{B}^{n}\) , with \(\mathcal{G}\) a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain ℝN-invariant model solutions. In case N+n=3, our 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 \(\mathcal{O}(t^{-1})\) and \(\mathcal{O}(t^{1/2})\) as t→∞.