We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on extremal Kerr black hole spacetimes. While the result was previously known for subextremal spacetimes, we show that the proof for the latter cannot be extended to the extremal case as the nature of the event horizon changes radically in the extremal limit.Finally, we explain how mode stability 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.

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