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The Stability Inequality for Ricci-Flat Cones

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Abstract

In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over ℂP 2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other examples of Ricci-flat cones over 4-manifolds are unstable, and that Ricci-flat cones over products of Einstein manifolds and over Kähler–Einstein manifolds with h 1,1>1 are unstable in dimension less than 10. As results of independent interest, our computations indicate that the Page metric and the Chen–LeBrun–Weber metric are unstable Ricci shrinkers. As a final bonus, we give plenty of motivations, and partly confirm a conjecture of Tom Ilmanen relating the λ-functional, the positive mass theorem, and the nonuniqueness of Ricci flow with conical initial data.

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Notes

  1. The C++ files are available on http://www.buckingham.ac.uk/directory/dr-stuart-hall/.

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Acknowledgements

We thank Richard Bamler, Pierre Germain, Nadine Große, Martin Li, Rafe Mazzeo, Thomas Murphy, Melanie Rupflin, Richard Schoen, Miles Simon, Michael Struwe, and especially Tom Ilmanen and Simon Donaldson for useful discussions and interesting comments. The research of RH and MS has been partly supported by the Swiss National Science Foundation; SH was supported by the EPSRC. RH also thanks the HIM Bonn for hospitality and financial support.

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Authors and Affiliations

  1. Department of Applied Computing, The University of Buckingham, Buckingham, UK

    Stuart Hall

  2. Courant Institute of Mathematical Sciences, New York University, New York, USA

    Robert Haslhofer

  3. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Siepmann

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  1. Stuart Hall
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  2. Robert Haslhofer
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  3. Michael Siepmann
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Correspondence to Robert Haslhofer.

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Hall, S., Haslhofer, R. & Siepmann, M. The Stability Inequality for Ricci-Flat Cones. J Geom Anal 24, 472–494 (2014). https://doi.org/10.1007/s12220-012-9343-z

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