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Following the work of Li-Shi-Qing, we propose the definition of the relativevolume function for an AH manifold. It is not a constant function in generaland we study the egularity of this function. We use this function to give anaccurate characterization of the height of the geodesic defining function forthe AH manifold with a given boundary metric. It is also proved that suchfunctions are uniformly bounded from below at infinity and the bound onlydepends on the dimension. As an application, we use this function to researchthe capacity of balls in AH manifold and provide some limit results.

The relative volume function and the capacity of sphere on asymptotically hyperbolic manifolds