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

TL;DR

This paper introduces the relative volume function for asymptotically hyperbolic manifolds, analyzes its regularity, and applies it to characterize geodesic defining functions and study the capacity of balls, revealing bounds and limit behaviors.

Contribution

It defines and studies the properties of the relative volume function on AH manifolds, providing new insights into their geometric structure and capacities.

Findings

The relative volume function is not constant in general.

It is uniformly bounded from below at infinity, depending only on the dimension.

The function helps characterize the height of geodesic defining functions and analyze capacities.

Abstract

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 egularity of this function. We use this function to give an accurate characterization of the height of the geodesic defining function for the AH manifold with a given boundary metric. It is also proved that such functions are uniformly bounded from below at infinity and the bound only depends on the dimension. As an application, we use this function to research the capacity of balls in AH manifold and provide some limit results.