Stable and unstable Einstein warped products

TL;DR

This paper investigates the stability of warped product Einstein manifolds, establishing conditions for stability based on eigenvalues and identifying classes of manifolds that are stable, including those with imaginary Killing spinors and certain cones.

Contribution

It provides a systematic characterization of stability for warped Einstein manifolds and identifies new stable classes such as manifolds with imaginary Killing spinors and specific Ricci-flat and hyperbolic cones.

Findings

All manifolds with imaginary Killing spinors are strictly stable.

Stability of Ricci-flat and hyperbolic cones over certain Kähler-Einstein Fano manifolds for dimensions ≥10.

Eigenvalue conditions characterize stability of these Einstein warped products.

Abstract

In this article, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killing spinors are strictly stable. Moreover, we show that Ricci-flat and hyperbolic cones over K\"ahler-Einstein Fano manifolds and over nonnegatively curved Einstein manifolds are stable if the cone has dimension $n\geq10$.