We examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic (n+1) -manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic submanifold. Our proof adapts the argument of Dyatlov and Jin to higher dimensions and classifies the closures of horocyclic orbits using Ratner theory. An important step in the proof is a generalization of the higher-dimensional fractal uncertainty principle of Cohen to Fourier integral operators, which may be of independent interest.
Kim et al. (2025) studied this question.