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Convergence and Stability of Locally ℝN-Invariant Solutions of Ricci Flow

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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→∞.

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  1. University of Texas at Austin, Austin, TX, USA

    Dan Knopf

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  1. Dan Knopf
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Correspondence to Dan Knopf.

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The author acknowledges NSF support in the form of grants DMS-0545984 and DMS-0505920.

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Knopf, D. Convergence and Stability of Locally ℝN-Invariant Solutions of Ricci Flow. J Geom Anal 19, 817–846 (2009). https://doi.org/10.1007/s12220-009-9091-x

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