Stability of Riemannian manifolds with Killing spinors

主讲教师:王常亮 人气:1067 更新时间: 2017年06月17日
摘要: Einstein metrics on a compact manifold are critical points of the normalized total scalar curvature functional. So it is natural to study the behavior of the second variation of the normalized total scalar curvature functional at an Einstein metric. This is known as the linear stability problem of Einstein metrics. Riemannian manifolds with non-zero Killing spinors are Einstein. We prove that complete manifolds with non-zero imaginary Killing spinors are stable by using a Bochner type formula, which was proved by McKenzie Wang and then was rediscovered by Xianzhe Dai, Xiaodong Wang, and Guofang Wei. This stability result has already been proved by Klaus Kroncke in a different way.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 circle bundle over a Kahler-Einstein manifold. We prove that if the base space is a product of two Kahler-Einstein manifolds then the regular Sasaki-Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors. More generally, we prove that Einstein metrics on principal torus bundles constructed by McKenzie Wang and Wolfgang Ziller are unstable if the base is a product of Kahler manifolds.
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