Question

I have an hyperbolic manifold with boundary a conformal sphere. Can I extend any conformal transformation of boundary to the interior of the ball? I know how to do with Moebius but I wonder if there is a general prescription for generic holomorphic transformations on the boundary

Answer 1

I assume you mean that you want to extend a holomorphic map $\phi:S^2\to S^2$ to a map $\Phi: \overline{\mathbb{H}^3}\to \overline{\mathbb{H}^3}$. Since a holomorphic map $\phi:S^2\to S^2$ is a rational map, and in particular a branched cover, you may extend it over the 3-ball (e.g. just by coning off). I assume, however, that you want to do it in some canonical way which is natural under conjugation by the Mobius group. For example, map $z\mapsto z^n$ gives a map of $S^2=\hat{\mathbb{C}}$ which extends in a natural way to $\mathbb{H}^3$ as a branched cover over a geodesic connecting $0$ and $\infty$ in $\mathbb{H}^3$. I think one may also be able to use the Douady-Earle extension as Igor suggests. The higher-dimensional version of this is given by Besson-Courtois-Gallot, and is called the "natural map". The rough idea is to associate to each point in $\mathbb{H}^3$ the visual measure on $\partial \mathbb{H}^3=S^2$, then push forward this measure by the map at infinity, and then take the barycenter of this measure to determine where the point goes. I don't see why this couldn't work for a rational map, although it may be tricky to compute. You might also have a look at a paper of Lyubich and Minsky who show how to associate a hyperbolic lamination to a rational map.

Answer 2

I dont really understand the question, but I suspect the keywords are "Douady-Earle extension"'; see

http://mathoverflow.net/questions/62924/quasiconformal-harmonic-extension-of-a-quasi-symmetric-map-on-s1

or the original paper:

http://www.springerlink.com/content/hw88761257310165/