Stability of Riemannian manifolds with Killing spinors

Abstract

Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^1$-bundle over a Kähler-Einstein manifold. We prove that if the base space is a product of two Kähler-Einstein manifolds then the regular Sasaki-Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.

Final version, introduction revised, references added

References in corpus (2)

• Bohm and Einstein-Sasaki Metrics, Black Holes and Cosmological Event Horizons
• Linear stability of Perelman's $\nu$-entropy on symmetric spaces of compact type