A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

We consider Kerr spacetimes with parameters a and M such that vertical bar a vertical bar << M, Kerr-Newman spacetimes with parameters vertical bar Q vertical bar << M, vertical bar a vertical bar << M and more generally, stationary axisymmetric black hole exterior spacetimes (M, g) which are sufficiently close to a Schwarzschild metric with parameter M > 0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation square(g)psi = 0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Sigma(tau), then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions psi, a suitable energy flux through Sigma(tau) is bounded by C times the initial energy flux through Sigma(0). This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of psi, in terms of a higher order initial energy on Sigma(0). Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.