Following the work of Li-Shi-Qing, we propose the definition of the relative volume function for an AH manifold. It is not a constant function in general and we study the regularity of this function. We use this function to provide an accurate characterization of the height of the geodesic defining function for the AH manifold with a given boundary metric. Furthermore, it is shown that such functions are uniformly bounded from below at infinity and the bound only depends on the dimension. In the end, we apply this function to study the capacity of balls in AH manifolds and demonstrate that the "relative $p$-capacity function" coincides with the relative volume function under appropriate curvature conditions.
Xiaoshang JIN
. THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS[J]. Acta mathematica scientia, Series B, 2025
, 45(3)
: 755
-770
.
DOI: 10.1007/s10473-025-0301-9
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