Let $\Sigma$ be a closed surface of genus at least $2$.
Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is identified with a domain $\Omega\subset\partial\mathbb{H}^3$, and the fundamental group $\Gamma$ identified with a subgroup of $\mathrm{PSL}(2,\mathbb{C})$.
By Donaldson's theorem on twisted harmonic maps, there is a unique $\Gamma$-equivariant harmonic map $\varphi:\Omega\rightarrow\mathbb{H}^3$. $\varphi(\Omega)$ is a surface in $\mathbb{H}^3$ with asymptotic boundary $\partial\Omega$. Assume that $\varphi$ is an immersion. Then there is a hyperbolic Gauss map $g$ sending $\varphi(\Omega)$ back to $\partial\mathbb{H}^3$. (To be precise, for each point $m$ on the surface $\varphi(\Omega)$, we let $v(m)\in T_m\mathbb{H}^3$ be the unit normal vector of the surface pointing towards $\Omega$. Then we follow the geodesic with initial velocity $v(m)$ to the infinity, the limit is defined to be $g(m)$.)
The map $g\circ \varphi: \Omega\rightarrow\partial\mathbb{H}^3$ is of course $\Gamma$-equivariant and is just the identity if the quasi-Fuchsian structure is actually Fuchsian.
Question: What can we say about $g\circ \varphi$ in general? Is it conformal? Is its image $\Omega$ itself? Is it always the identity?
