Capacity of Balls in Hyperbolic Space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:29:34Z http://mathoverflow.net/feeds/question/60590 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60590/capacity-of-balls-in-hyperbolic-space Capacity of Balls in Hyperbolic Space ght 2011-04-04T18:49:26Z 2011-04-06T04:40:16Z <p>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 $\partial\Omega$. </p> <blockquote> <p>Is it known what is the capacity of a ball of radius $r$ in the $n$-th dimensional hyperbolic space?</p> </blockquote> http://mathoverflow.net/questions/60590/capacity-of-balls-in-hyperbolic-space/60608#60608 Answer by Anton Petrunin for Capacity of Balls in Hyperbolic Space Anton Petrunin 2011-04-04T20:51:49Z 2011-04-06T04:40:16Z <p>Sounds like a home work; here are some hints:</p> <p>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 $$\mathrm{vol}\, S^{n-1}{\cdot}\int\limits_R^\infty(f')^2{\cdot}(\sinh r)^{n-1}\, dr.$$</p> <p>(I do not know if will get the answer in a simple form, but it will be good for all practical purposes...)</p> http://mathoverflow.net/questions/60590/capacity-of-balls-in-hyperbolic-space/60683#60683 Answer by ght for Capacity of Balls in Hyperbolic Space ght 2011-04-05T13:13:40Z 2011-04-05T13:13:40Z <p>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 $\Omega$.</p> <p>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.</p> <p>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 $$\mathrm{cap}(B(x,r))=-\mathrm{vol}(S^{n-1})f'(r)$$</p> <p>I'm in a rush I hope it makes sense!</p>