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On the long-time behavior of immortal Ricci flows

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Abstract

For an immortal Ricci flow on an m-dimensional \((m\ge 3)\) closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a Riemannian orbifold; (2) if the flow is type-III with diameter growth controlled by \(t^{\frac{1}{2}}\), then any blowdown limit is an m-dimensional negative Einstein manifold, provided that Feldman–Ilmanen–Ni’s \(\varvec{\mu }_+\)-functional satisfies \(\lim _{t\rightarrow \infty } t\varvec{\mu }_+'(t)=0\).

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Acknowledgements

I would like to thank Yu Li, Xiaochun Rong, Bing Wang and Ruobing Zhang for useful discussions during the preparation of the paper. I would also like to thank Richard Bamler, John Lott, Aaron Naber and Song Sun for helpful comments on the non-collapsing and Ricci flatness of the long-time limits by the evolution of immortal Ricci flows with uniformly bounded curvature and diameter, as well as on the case of comapct type-III Ricci flows. Finally, I would like to thank an anonymous referee for valuable comments on the paper.

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  1. Department of Mathematics, University of Wisconsin - Madison, 480 Lincoln Drive, Madison, WI, 53706, USA

    Shaosai Huang

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Correspondence to Shaosai Huang.

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Communicated by A. Chang.

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Huang, S. On the long-time behavior of immortal Ricci flows. Calc. Var. 60, 78 (2021). https://doi.org/10.1007/s00526-021-01941-7

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