Question

Let $\Gamma\subset PSL(2,R)$ be a Fuchsian group. For which representations $\rho:\Gamma\to PSL(2,R)$ does there exist a harmonic map from the hyperbolic plane to itself satisfying

$f(\gamma z)=\rho(\gamma)f(z)$

for all $\gamma\in \Gamma$?

I was told that when $\Gamma$ is co-compact and torsion free, the existence of such an $f$ was given by Simpson's correspondence. I am interested to know if this extends in any way to more general $\Gamma$, especially co-finte volume $\Gamma$, and if this fact can be deduced in an elementary way.

Answer 1

The cocompact case in this setting is due to Donaldson: S.Donaldson, Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) 55 (1987), no. 1, 127–131.

Simpson proved a general theorem for more general target groups and when the domain is a compact Kahler manifold.

For non-cocompact groups, I presume, you are interested in finite-energy harmonic maps. For lattices $\Gamma\subset PSL(2, {\mathbb R})$, see for instance the paper by Jost, Yang, Zuo: http://arxiv.org/pdf/math/0505144v1.pdf (appeared in J. Reine Angew. Math. 609 (2007), 137--159) where more general target spaces are discussed as well.