Given a non-constant holomorphic quadratic differential $\phi \, \mathrm{d} z^2$ on the complex plane $\mathbf{C}$, there is a harmonic diffeomorphism $u: \mathbf{C} \to \mathbf{H}$ into the hyperbolic plane, whose Hopf differential is $\Phi(u) = \phi \, \mathrm{d} z^2$.
This was a result proved in 1994 by Wan and Au [WA94]. The proof is obtained by rewriting the harmonic map equation as the PDE $\Delta w = \mathrm{e}^{2w} - \lvert \phi \rvert^2 \mathrm{e}^{-2w}$, and then constructing super-and subsolutions for this.
I am trying to get a better grasp of these harmonic maps. Is there a good description for example for polynomial differentials, with $\phi$ having roots at $a_1,\dots,a_d \in \mathbf{C}$ say?
- I think I'd be perfectly content with a description of the harmonic maps that correspond to the Hopf differentials of the form $z^k \, \mathrm{d} z^2$, where $k \geq 1$ is integer.
- I am happy to read up on this myself, if you have a suggestion for a good reference!
[WA94] Wan and Au. Parabolic constant mean curvature spacelike surfaces. Proc. AMS. Vol 120, (2), 1994.
