A robust measure of decay and dispersion for the wave equation is provided by the localized energy estimates, which have been essential in proving, e.g. the Strichartz estimates on black hole backgrounds. We study localized energy estimates for the wave equation on (1+4)-dimensional Myers-Perry space-times, which represent a family of rotating asymptotically at black holes with spherical horizon topology and generalize the well-known Kerr space-times to higher dimensions. Because of the extra dimension, the Myers-Perry family is parameterized by two angular momentum parameters, which we assume to be sufficiently small relative to the mass of the black hole, essentially allowing us to treat the space-time as a perturbation of the Schwarzschild black hole. This investigation is motivated by the nonlinear stability problem for the Kerr family of black holes, which may be easier to understand in higher dimensions. Typically, the localized energy estimates are proved by commuting the wave operator with a suitable first-order differential operator and integrating by parts. However, the underlying black hole geometry introduces a number of difficulties related to the trapping phenomenon, which is a known obstruction to dispersion and necessitates a loss in decay. This phenomenon is manifest along the event horizon of the Schwarzschild/Kerr black holes, but its effect is rendered negligible due to the celebrated red-shift effect. More delicate analysis is required to deal with trapping that occurs along e.g., the so-called photon sphere in the Schwarzschild geometry. Localized energy estimates on higher dimensional Schwarzschild black holes were proved by Laul-Metcalfe in [34] using a single differential multiplier, but their method relies fundamentally on the fact that the trapped null geodesics lie on a sphere. On the Myers-Perry space-time, the nature of the trapped set is much more complicated and must be described in phase space rather than by position alone, and consequently a single differential multiplier is insufficient to prove the desired result. Once it is determined that all trapped geodesics lie on surfaces of constant r, we can adapt the method of Tataru and Tohaneanu in [62], which perturbs off the Schwarzschild case by instead commuting with an appropriate pseudodifferential operator to generate a positive commutator near the trapped set. This describes joint work with Parul Laul, Jason Metcalfe, and Mihai Tohaneanu [35].Bachelor of Scienc
