Capacity of Balls in Hyperbolic Space 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?

 A: Sounds like a home work; here are some hints:
The capacity is equal to the integral of |gradient|$^2$ of spherically symmetric harmonic function
with 1 on the bry of the ball and zero at infinity.
The function $f$ depends only on the radius, say $r$.
You can cook an ODE for $f$, something like 
$$f''(r)+\frac{(n-1){\cdot}\cosh r}{\sinh r}{\cdot}f'(r)=0.$$
Then you should solve it and integrate
$$\mathrm{vol}\\, S^{n-1}{\cdot}\int\limits_R^\infty(f')^2{\cdot}(\sinh r)^{n-1}\\, dr.$$
(I do not know if will get the answer in a simple form, but it will be good for all practical purposes...)
A: 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!
