The Stability Inequality for Ricci-Flat Cones

Abstract

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.

On the spectrum of the Page and the Chen–LeBrun–Weber metrics

“…In the recent paper [14] the first author, Robert Haslhofer and Michael Siepmann gave an alternative proof of the instability of the Page metric based on the presence of many (> 1) harmonic 2-forms on this manifold. There the Bunch-Donaldson numerical approximation to the Chen-LeBrun-Weber metric [5] was used to give strong evidence that the Chen-LeBrun-Weber metric is also unstable.…”

Stable and unstable Einstein warped products

“…On the other hand, as was pointed out in [HHS14], −2(n−1) ∈ spec(∆ E | T T ) if (M, g) is a product of positive Einstein manifolds or if (M, g) is a positive Kähler-Einstein manifolds with dim(H 1,1 (M )) > 1. Thus, we obtain Theorem 4.6 ([HHS14, Theorem 1.1 and Theorem 1.2]).…”

“…The main theorems of this paper are stability theorems about these models. Ricci-flat cones have already been considered in [HHS14] and we partly build up on that work. .…”

A mass-decreasing flow in dimension three

“…[7,32,29,17,16,14] for more information on this aspect. Finally, the Perelmanenergy and its variantλ = sup λV 2/3 numerically characterize the long-time behavior of the Ricci flow with surgery on closed three-manifolds [27].…”