The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves

The geometry of solutions to the higher dimensional Einstein vacuum equationspresents aspects that are absent in four dimensions, one of the most remarkablebeing the existence of stably trapped null geodesics in the exterior ofasymptotically flat black holes. This paper investigates the stable trappingphenomenon for two families of higher dimensional black holes, namely blackstrings and black rings, and how this trapping structure is responsible for theslow decay of linear waves on their exterior. More precisely, we study decayproperties for the energy of solutions to the scalar, linear wave equation$\Box_{g_{\textup{ring}}} \Psi=0$, where $g_{\textup{ring}}$ is the metric of afixed black ring solution to the five-dimensional Einstein vacuum equations.For a class $\mathfrak{g}$ of black ring metrics, we prove a logarithmic lowerbound for the uniform energy decay rate on the black ring exterior$(\mathcal{D},g_{\textup{ring}})$, with $g_{\textup{ring}}\in\mathfrak{g}$. Theproof generalizes the perturbation argument and quasimode construction ofHolzegel--Smulevici \cite{SharpLogHolz} to the case of a non-separable waveequation and crucially relies on the presence of stably trapped null geodesicson $\mathcal{D}$. As a by-product, the same logarithmic lower bound can beestablished for any five-dimensional black string. Our result is the first mathematically rigorous statement supporting theexpectation that black rings are dynamically unstable to generic perturbations.In particular, we conjecture a new \textit{nonlinear} instability forfive-dimensional black strings and thin black rings which is already present atthe level of scalar perturbations and clearly differs from the mechanism drivenby the well-known Gregory--Laflamme instability.