The stability inequality for Ricci-flat cones

Hall, Stuart J. and Haslhofer, Robert and Seipmann, Michael (2014) The stability inequality for Ricci-flat cones. Journal of Geometric Analysis, 24 (1). pp. 472-494. ISSN 1050-6926 (Print) 1559-002X (Online)

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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 lambda-functional, the positive mass theorem and the nonuniqueness of Ricci flow with conical initial data.

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