The stability inequality for Ricci-flat cones

TL;DR

This paper investigates the stability of Ricci-flat cones, proving the stability of the cone over CP^2 and instability of many others, with implications for Ricci flow and geometric analysis.

Contribution

It establishes the first stable non-flat Ricci-flat cone in minimal dimension and analyzes stability properties of various Ricci-flat cones over complex and Einstein manifolds.

Findings

Ricci-flat cone over CP^2 is stable

Many Ricci-flat cones over 4-manifolds are unstable

Ricci-flat cones over certain Einstein manifolds are unstable in dimensions less than 10

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 Kahler-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.