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Stability and instability of Ricci solitons

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Abstract

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton \((M,g)\) is a local maximum of Perelman’s shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If \(g\) is not a local maximum of the shrinker entropy, we show that there exists a nontrivial normalized Ricci flow emerging from it. These theorems are analogues of results in the Ricci-flat and in the Einstein case (Haslhofer and Müller, arXiv:1301.3219, 2013; Kröncke, arXiv:1312.2224, 2013).

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Acknowledgments

The author would like to thank Christian Bär, Stuart Hall and Thomas Murphy for their support and helpful discussions. Moreover, the author thanks Sonderforschungsbereich 647 funded by Deutsche Forschungsgemeinschaft for financial support.

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  1. Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469,  Potsdam, Germany

    Klaus Kröncke

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  1. Klaus Kröncke
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Correspondence to Klaus Kröncke.

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Communicated by M. Struwe.

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Kröncke, K. Stability and instability of Ricci solitons. Calc. Var. 53, 265–287 (2015). https://doi.org/10.1007/s00526-014-0748-3

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