具有有界Killing向量场的引力瞬子的稳定性
Stability of gravitational instantons with a bounded Killing vector field
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中文总结 AI 辅助
该研究证明带有有界Killing向量场的完备ALF Ricci平坦4-流形线性稳定当且仅当局部超凯勒，还给出高维AF流形推广的稳定性结论及相关度量的去稳定张量。
中文摘要 AI 辅助
我们证明，带有有界Killing向量场的完备ALF Ricci平坦4-流形是线性稳定的，当且仅当它是局部超凯勒的。这一结论适用于所有已知的例子，尤其证明了Li-Sun近期发现的所有度量的不稳定性。该论证基于与Killing场相关的Ricci平坦度量的无穷小Einstein-Maxwell形变。一个独立的更简单的恒等式证明了非平坦静态Ricci平坦4-度量的不稳定性，其中包括静态轴对称光滑黎曼Myers/Korotkin-Nicolai度量。据我们所知，这一结果证明了所有已知Ricci平坦4-流形族的线性稳定性与特殊 holonomy（和反）自对偶性之间的等价性。在高维情形下，对于AF流形的推广，我们证明稳定性迫使万有覆盖分裂出一条直线。这为所有维度下的黎曼Myers-Perry瞬子和黎曼Schwarzschild-Tangherlini度量提供了显式的去稳定张量。
英文摘要
We prove that a complete ALF Ricci-flat 4-manifold carrying a bounded Killing vector field is linearly stable if and only if it is locally hyperkähler. This applies uniformly to all known examples and in particular proves the instability of all of the metrics recently found by Li-Sun. The argument is based on infinitesimal Einstein-Maxwell deformations of the Ricci-flat metric associated with the Killing field. A separate simpler identity proves the instability of nonflat static Ricci-flat 4-metrics which include the static axisymmetric smooth Riemannian Myers/Korotkin-Nicolai metrics. To our knowledge, this proves the equivalence between linear stability and special holonomy and (anti-)selfduality for all known families of Ricci-flat 4-manifolds. In higher dimensions, for generalizations of AF manifolds, we show that stability forces the universal cover to split a line. This gives explicit destabilizing tensors on the Riemannian Myers-Perry instantons and the Riemannian Schwarzschild-Tangherlini metrics in all dimensions.
