Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int\_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ ranges over all continuous, compactly supported functions on $M\setminus\Omega$ which are $C^{\infty}$ on $M\setminus\overline{\Omega}$ and which are equal to 1 on $\partial\Omega$. 



> 
> Is it known what is the capacity of a
>  ball of radius $r$ in the
>  $n$-th dimensional hyperbolic space?
> 
> 
> 



Capacity of Balls in Hyperbolic Space
The capacity of a set $\Omega$ it is known to be
$$
\mathrm{cap}(\Omega)=-\int\_{\partial\Omega}{\frac{\partial \Phi}{\partial \nu}dA}
$$
where $\Phi$ is an harmonic function with $\Phi|\partial\Omega=1$ and $A$ is the $(n-1)$ area of $\partial\Omega$ and $\frac{\partial}{\partial\nu}$ is the normal derivative along $\partial\Omega$ exterior to $\Omega$.


The Laplacian in the hyperbolic space in polar coordinates has the form:
$$
\Delta\_{H^{n}} f(t,\xi) = \sinh^{1-n}t \frac{\partial}{\partial t}\left(\sinh^{n-1}t\frac{\partial f}{\partial t}\right) + \sinh^{-2}t\Delta\_\xi f 
$$
where $\Delta\xi$ is the Laplace–Beltrami operator on the ordinary unit $n-1$-sphere.


Therefore, if $f$ is a radial harmonic function then:
$$
\Delta\_{H^{n}}(f)=0 \iff (n-1)\cosh(t)f'(t)+\sinh(t)f''(t)=0
$$
and 
$$
\mathrm{cap}(B(x,r))=-\mathrm{vol}(S^{n-1})f'(r)
$$


I'm in a rush I hope it makes sense!


