Harmonic Maps to Teichmueller Space

G. Daskalopoulos, L. Katzarkov, and R. Wentworth

We give sufficient conditions for the existence of equivariant harmonic maps from the universal cover of a Riemann surface $B$ to the Teichm\"uller space of a genus $g\geq 2$ surface $\Sigma$. The condition is in terms of the representation of the fundamental group of $B$ to the mapping class group of $\Sigma$. The metric on Teichm\"uller space is chosen to be the K\"ahler hyperbolic metric. Examples of such representations arise from symplectic Lefschetz fibrations.