# Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$.

If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial X$, then by Korevaar-Schoen (or Corlette-Donaldson, Labourie) there exists a $\rho$-equivariant harmonic map $$H^3=\widetilde{M}\rightarrow X$$

What can be said about the induced boundary map between the geometric boundaries $$S^2=\partial H^3\rightarrow \partial X?$$ Is it quasiconformal? injective? an immersion?

• Measurable and nothing else in general. – Misha Oct 24 '13 at 21:18

I will assume, in addition, that $M$ is compact and that the target space is again ${\mathbb H}^n$, $n\ge 1$. Now, here are some examples to ponder:
1. Suppose that the image of $\rho$ is a Schottky group (the limit set is a Cantor set $C$), then the boundary map $f$ sends $S^2$ to $C$. Can such a map be continuous?
2. The image of $\rho$ is a dense subgroup of $Isom({\mathbb H}^n)$. Can the map $f$ be continuous in this case?
Thinking about such examples will help you to appreciate how complicated the boundary map $f$ is. In general, this map is only measurable. Sometimes, you can get a better conclusion. For instance, if $X$ is Gromov-hyperbolic and $\rho$ is an isomorphism to a quasiconvex isometry group, then the boundary map will be quasisymmetric. However, this has nothing to do with $h: {\mathbb H}^3\to X$ being harmonic, all you need is equivariance.