4 problems

Define the bounded-motion region … where $R$ is the Kerr radial potential, $r^*$ is the local maximum of $-R$, and $r_*$ is the local minimum of $-R$ to its right. Generic Kerr bou…

Let $N_\varepsilon$ be the perturbed normally hyperbolic invariant manifold associated with the Kerr photon shell, and let $(a,M,E,Q,L_z)$ satisfy the conditions in assumption (2)…

Define the bounded-motion region … Here $E$ is the particle energy, $L$ its angular momentum, $V$ the effective potential, and $H$ the Hamiltonian. Generic Schwarzschild bounded-ph…

Let $N_\varepsilon$ denote the perturbed normally hyperbolic invariant manifold associated with the photon sphere, and let $L$ and $L_z$ be the angular-momentum quantities used in…