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

Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold