We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on Kerr black hole spacetimes, both in the extremal and subextremal case. We also give a quantitative refinement of mode stability. As an immediate application, we show that the transmission and reflection coefficients of the scattering problem are bounded, independently of the specific angular momentum of the black hole, in any compact set of real frequencies excluding zero frequency and the superradiant threshold. While in the subextremal setting these results were known previously, the extremal case is more involved and has remained an open problem. Ours are the first results outside axisymmetry and could serve as a preliminary step towards understanding boundedness, scattering and decay properties of general solutions to the Teukolsky equation on extremal Kerr black holes.

中文翻译：

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Teukolsky 方程在极值和次极值克尔时空上的模式稳定性

我们证明，在极值和次极值情况下，克尔黑洞时空上的 Teukolsky 方程的实轴上没有指数增长模式或模式。我们还对模式稳定性进行了定量改进。作为一个直接的应用，我们证明了散射问题的透射和反射系数是有界的，与黑洞的特定角动量无关，在除零频率和超辐射阈值之外的任何紧凑实频率集合中。虽然在亚极值环境中，这些结果以前是已知的，但极值情况涉及更多，并且仍然是一个悬而未决的问题。我们的结果是轴对称之外的第一个结果，可以作为理解有界性的初步步骤，