Mode stabilities and instabilities for scalar fields on Kerr exterior spacetimes

Abstract

In this thesis we study wave and Klein-Gordon equations on Kerr exterior spacetimes. For the wave equation, we give a quantitative refinement and simple proofs of mode stability type statements on Kerr backgrounds in the full sub-extremal range ([absolute value of]a < M). As an application, we are able to quantitatively control the energy flux along the horizon for solutions to the wave equation in any bounded-frequency regime. This estimate plays a crucial role in the author's recent proof, joint with Mihalis Dafermos and Igor Rodnianski, of boundedness and decay for the solutions to the wave equation on the full range of sub-extremal Kerr spacetimes. For the Klein-Gordon equation, we show that given any Kerr exterior spacetime with non-zero angular momentum, we may find an open family of non-zero Klein-Gordon masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein-Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to [absolute value of]am/2Mr+.