Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper, we generalize this result to asymptotically hyperbolic (AH) Einstein manifold. We get an upper bound of the relative volume of AH manifold and if we combine it with the recent work of Wang and Zhou, then the rigidity is obtained.