正弦锥和双曲柱面的稳定性
Stability of sin-cones and cosh-cylinders
摘要：
这项工作涉及的稳定性和不稳定性的爱因斯坦翘曲产品与爱因斯坦纤维的余维1。我们研究的情况下，数量曲率的翘曲产品和纤维要么都是积极的或都是消极的，以补充[Krö16]中的结果。在正弦锥的情况下，直到一个小的间隙，这种扭曲的产品的稳定性现在完全由光纤的拉普拉斯算子和爱因斯坦算子的光谱特性决定。对于双曲余弦圆柱，我们还能够证明一个收敛结果的里奇流开始在一个小的邻域。作为一类有趣的例子，我们确定了紧型对称空间上的正弦锥的稳定性。
英文摘要：
This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Krö16]. Up to a small gap in the case of sin-cones, the stability properties of such warped products are now completely determined by spectral properties of the Laplacian and the Einstein operator of the fiber. For cosh-cylinders, we are furthermore able to prove a convergence result for the Ricci flow starting in a small neighbourhood. As an interesting class of examples, we determine the stability properties of sin-cones over symmetric spaces of compact type.
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