General conformal transformation extension to hyperbolic manifold I have a hyperbolic manifold with boundary a conformal sphere. Can I extend any conformal transformation of the 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.
 A: 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 Möbius 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 Entropies et rigidités des espaces localement symétriques de courbure strictement négative, 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, Laminations in holomorphic dynamics who show how to associate a hyperbolic lamination to a rational map.
A: I don't really understand the question, but I suspect the keywords are "Douady–Earle extension"; see the question Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$, or the original paper:

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*Adrien Douady, Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157: 23-48 (1986). doi:10.1007/BF02392590
